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## Material Information- Title:
- Pressure-based methods on single-instruction stream/multiple-data stream computers
- Creator:
- Blosch, Edwin L
- Publication Date:
- 1994
- Language:
- English
- Physical Description:
- vii, 189 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Algorithms ( jstor )
Cavity flow ( jstor ) Convection ( jstor ) Data smoothing ( jstor ) Mathematical procedures ( jstor ) Multigrid methods ( jstor ) Perceptron convergence procedure ( jstor ) Run time ( jstor ) Truncation errors ( jstor ) Velocity ( jstor ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1994.
- Bibliography:
- Includes bibliographical references (leaves 182-188).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- Edwin L. Blosch.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 021584257 ( ALEPH )
AKN2725 ( NOTIS ) 33373637 ( OCLC )
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PRESSURE-BASED METHODS ON SINGLE-INSTRUCTION STREAM/MULTIPLE-DATA STREAM COMPUTERS By EDWIN L. BLOSCH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 ACKNOWLEDGEMENTS I would like to express my thanks to my advisor Dr. Wei Shyy for reflecting carefully on my results and for directing my research toward interesting issues. I would also like to thank him for the exceptional personal support and flexibility he offered me during my last year of study, which was done off-campus. I would also like to acknowledge the contributions of the other members of my Ph.D. committee, Dr. Chen-Chi Hsu, Dr. Bruce Carroll, Dr. David Mikolaitis, and Dr. Sartaj Sahni. Dr. Hsu and Dr. Carroll supervised my B.S. and M.S. degree research studies, respectively, and Dr. Mikolaitis, in the role of graduate coordinator, enabled me to obtain financial support from the Department of Energy. Also I would like to thank Madhukar Rao, Rick Smith and H.S. Udaykumar, for paying fees on my behalf and for registering me for classes while I was in California. Jeff Wright, S. Thakur, Shin-Jye Liang, Guobao Guo and Pedro Lopez-Fernandez have also made direct and indirect contributions for which I am grateful. Special thanks go to Dr. Jamie Sethian, Dr. Alexandre Chorin and Dr. Paul Con- cus of Lawrence Berkeley Laboratory for allowing me to visit LBL and use their resources, for giving personal words of support and constructive advice, and for the privilege of interacting with them and their graduate students in the applied mathe- matics branch. Last but not least I would like to thank my wife, Laura, for her patience, her example, and her frank thoughts on "cups with sliding lids," "flow through straws," and numerical simulations in general. My research was supported in part by the Computational Science Graduate Fel- lowship Program of the Office of Scientific Computing in the Department of Energy. The CM-5s used in this study were partially funded by National Science Foundation Infrastructure Grant CDA-8722788 (in the computer science department of the Uni- versity of California-Berkeley), and a grant of HPC time from the DoD HPC Shared Resource Center, Army High-Performance Computing Research Center, Minneapolis, Minnesota. TABLE OF CONTENTS ACKNOWLEDGEMENTS ............................ ii ABSTRACT ..................... ............... vi CHAPTERS 1 INTRODUCTION ..................... .......... 1 1.1 Motivations ............................... 1 1.2 Governing Equations ........................... 3 1.3 Numerical Methods for Viscous Incompressible Flow .......... 5 1.4 Parallel Com putting ............................ 7 1.4.1 Data-Parallelism and SIMD Computers ................ 8 1.4.2 Algorithms and Performance ..................... 11 1.5 Pressure-Based Multigrid Methods .... 13 1.6 Description of the Research ..... ..... 17 2 PRESSURE-CORRECTION METHODS .... 21 2.1 Finite-Volume Discretization on Staggered Grids .... 21 2.2 The SIMPLE Method ................. ....... 23 2.3 Discrete Formulation of the Pressure-Correction Equation ...... 27 2.4 Well-Posedness of the Pressure-Correction Equation ... 30 2.4.1 Analysis ... .. . 30 2.4.2 Verification by Numerical Experiments .... 33 2.5 Numerical Treatment of Outflow Boundaries .... 38 2.6 Concluding Remarks ................... ........ 40 3 EFFICIENCY AND SCALABILITY ON SIMD COMPUTERS ...... 53 3.1 Background ............... .. .............. 53 3.1.1 Speedup and Efficiency .. .. 53 3.1.2 Comparison Between CM-2, CM-5, and MP-1 ... 55 3.1.3 Hierarchical and Cut-and-Stack Data Mappings ... 57 3.2 Implementional Considerations .. ... 59 3.3 Numerical Experiments ... 61 3.3.1 Efficiency of Point and Line Solvers for the Inner Iterations 62 3.3.2 Effect of Uniform Boundary Condition Implementation 69 3.3.3 Overall Performance ....................... 70 3.3.4 Isoefficiency Plot ......................... 72 3.4 Concluding Remarks ........................... 4 A NONLINEAR PRESSURE-CORRECTION MULTIGRID METHOD .. 4.1 Background . . 4.1.1 Terminology and Scheme for Linear Equations ......... 4.1.2 Full-Approximation Storage Scheme for Nonlinear Equations 4.1.3 Extension to the Navier-Stokes Equations . 4.2 Comparison of Pressure-Based Smoothers . 4.3 Stability of Multigrid Iterations . . 4.3.1 Defect-Correction Method . .. 4.3.2 Cost of Different Convection Schemes ..... .. 4.4 Restriction and Prolongation Procedures . 4.5 Concluding Remarks ........................... 5 IMPLEMENTATION AND PERFORMANCE ON THE CM-5 ....... 5.1 Storage Problem .......... ..... .... ....... ... 5.2 Multigrid Convergence Rate and Stability . 5.2.1 Truncation Error Convergence Criterion for Coarse Grids . 5.2.2 Numerical Characteristics of the FMG Procedure . 5.2.3 Influence of Initial Guess on Convergence Rate . 5.2.4 Rem arks . . 5.3 Performance on the CM-5 ........................ 5.4 Concluding Remarks .......................... REFERENCES ................................... BIOGRAPHICAL SKETCH ............................ Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRESSURE-BASED METHODS ON SINGLE-INSTRUCTION STREAM/MULTIPLE-DATA STREAM COMPUTERS By Edwin L. Blosch Chairman: Dr. Wei Shyy Major Department: Aerospace Engineering, Mechanics and Engineering Science Computationally and numerically scalable algorithms are needed to exploit emerg- ing parallel-computing capabilities. In this work pressure-based algorithms which solve the two-dimensional incompressible Navier-Stokes equations are developed for single-instruction stream/multiple-data stream (SIMD) computers. The implications of the continuity constraint for the proper numerical treatment of open boundary problems are investigated. Mass must be conserved globally so that the system of linear algebraic pressure-correction equations is numerically consistent. The convergence rate is poor unless global mass conservation is enforced explicitly. Using an additive-correction technique to restore global mass conservation, flows which have recirculating zones across the open boundary can be simulated. The performance of the single-grid algorithm is assessed on three massively- parallel computers, MasPar's MP-1 and Thinking Machines' CM-2 and CM-5. Paral- lel efficiencies approaching 0.8 are possible with speeds exceeding that of traditional vector supercomputers. The following issues relevant to the variation of parallel ef- ficiency with problem size are studied: the suitability of the algorithm for SIMD computation; the implementation of boundary conditions to avoid idle processors; vi the choice of point versus line-iterative relaxation schemes; the relative costs of the coefficient computations and solving operations, and the variation of these costs with problem size; the effect of the data-array-to-processor mapping; and the relative speeds of computation and communication of the computer. A nonlinear pressure-correction multigrid algorithm which has better convergence rate characteristics than the single-grid method is formulated and implemented on the CM-5. On the CM-5, the components of the multigrid algorithm are tested over a range of problem sizes. The smoothing step is the dominant cost. Pressure-correction methods and the locally-coupled explicit method are equally efficient on the CM-5. V cycling is found to be much cheaper than W cycling, and a truncation-error based "full-multigrid" procedure is found to be a computationally efficient and convenient method for obtaining the initial fine-grid guess. The findings presented enable further development of efficient, scalable pressure-based parallel computing algorithms. CHAPTER 1 INTRODUCTION 1.1 Motivations Computational fluid dynamics (CFD) is a growing field which brings together high-performance computing, physical science, and engineering technology. The dis- tinctions between CFD and other fields such as computational physics and computa- tional chemistry are largely semantic now, because increasingly more interdisplinary applications are coming within range of the computational capabilities. CFD algo- rithms and techniques are mature enough that the focus of research is expected to shift in the next decade toward the development of robust flow codes, and toward the application of these codes to numerical simulations which do not idealize either the physics or the geometry and which take full account of the coupling between fluid dynamics and other areas of physics [65]. These applications will require formidable resources, particularly in the areas of computing speed, memory, storage, and in- put/output bandwidth [78]. At the present time, the computational demands of the applications are still at least two orders-of-magnitude beyond the computing technology. For example, NASA's grand challenges for the 1990s are to achieve the capability to simulate vis- cous, compressible flows with two-equation turbulence modelling over entire aircraft configurations, and to couple the fluid dynamics simulation with the propulsion and aircraft control systems modelling. To meet this challenge it is estimated that 1 ter- aflops computing speed and 50 gigawords of memory will be required [24]. Current massively-parallel supercomputers, for example, the CM-5 manufactured by Thinking Machines, have peak speeds of 0(10 gigaflops) and memories of 0(1 gigaword). Optimism is sometimes circulated that teraflop computers may be expected by 1995 [68]. In view of the two orders-of-magnitude disparity between the speed of present-generation parallel computers and teraflops, such optimism should be dimmed somewhat. Expectations are not being met in part because the applications, which are the driving force behind the progress in hardware, have been slow to develop. The numerical algorithms which have seen two decades of development on traditional vec- tor supercomputers are not always easy targets for efficient parallel implementation. Better understanding of the basic concepts and more experience with the present generation of parallel computers is a prerequisite for improved algorithms and imple- mentations. The motivation of the present work has been the opportunity to investigate issues related to the use of parallel computers in CFD, with the hope that the knowledge gained can assist the transition to the new computing technology. The context of the research is the numerical solution of the 2-d incompressible Navier-Stokes equations, by a popular and proven numerical method known as the pressure-correction tech- nique. A specific objective emerged as the research progressed, namely to develop and analyze the performance of pressure-correction methods on the single-instruction stream/multiple-data stream (SIMD) type of parallel computer. Single-grid compu- tations were studied first, then a multigrid method was developed and tested. SIMD computers were chosen because they are easier to program than multiple- instruction stream/multiple-data stream (MIMD) computers explicitt message-passing is not required), because synchronization of the processors is not an issue, and be- cause the factors affecting the parallel run time and computational efficiency are easier to identify and quantify. Also, these are arguably the most powerful machines available right now-Los Alamos National Laboratory has a 1024-node CM-5 with 32 Gbytes of processor memory and is capable of 32 Gflops peak speed. Thus, the code, the numerical techniques, and the understanding which are the contribution of this research can be immediately useful for applications on massively parallel computers. 1.2 Governing Equations The governing equations for 2-d, constant property, time-dependent viscous in- compressible flow are the Navier-Stokes equations. They express the principles of conservation of mass and momentum. In primitive variables and cartesian coordi- nates, they may be written u+ = 0 (1.1) apu apu2 apuv op a2u a2u + + = + 2 + y (1.2) dpv apuv dpv2 dp d2v a2v -+ +$- = -- (1.3) -t + + y dy +d2 + y2 where u and v are cartesian velocity components, p is the density, p is the fluid's molecular viscosity, and p is the pressure. Eq. 1.1 is the mass continuity equation, also known as the divergence-free constraint since its coordinate-free form is div ii = 0. The Navier-Stokes equations 1.1-1.3 are a coupled set of nonlinear partial differ- ential equations of mixed elliptic/parabolic type. Mathematically, they differ from the compressible Navier-Stokes equations in two important respects that lead to dif- ficulties for devising numerical solution techniques. First, the role of the continuity equation is different in incompressible flow. In- stead of a time-dependent equation for the density, in incompressible fluids the conti- nuity equation is a constraint on the admissible velocity solutions. Numerical meth- ods must be able to integrate the momentum equations forward in time while simul- taneously maintaining satisfaction of the continuity constraint. On the other hand, numerical methods for compressible flows can take advantage of the fact that in the unsteady form each equation has a time-dependent term. The equations are cast in vector form-any suitable method for time-integration can be employed on the system of equations as a whole. The second problem, assuming that a primitive-variable formulation is desired, is that there is no equation for pressure. For compressible flows, the pressure can be de- termined from the equation of state of the fluid. For incompressible flow, an auxiliary "pressure-Poisson" equation can be derived by taking the divergence of the vector form of the momentum equations; the continuity equation is invoked to eliminate the unsteady term in the result. The formulation of the pressure-Poisson equation requires manipulating the discrete forms of the momentum and continuity equations. A particular discretization of the Laplacian operator is therefore implied in pressure- Poisson equation, depending on the discrete gradient and divergence operators. This operator may not be implementable at boundaries, and solvability constraints can be violated [30]. Also, the differentiation of the governing equations introduces the need for additional unphysical boundary conditions on the pressure. Physically, the pressure in incompressible flow is only defined relative to an (arbitrary) constant. Thus, the correct boundary conditions are Neumann. However, if the problem has an open boundary, the governing equations should be supplemented with a boundary condition on the normal traction [29, 32], 1 &un F, -p + (1.4) Re dn where F is the force, Re is the Reynolds number, and the subscript n indicates the normal direction. However, F, may be difficult to prescribe. In practice, a zero-gradient or linear extrapolation for the normal velocity com- ponent is a more popular outflow boundary condition. Many outflow boundary con- ditions have been analyzed theoretically for incompressible flow (see [30, 31, 38, 56]). There are even more boundary condition procedures in use. The method used and its impact on the "solvability" of the resulting numerical systems of equations depends on the discretization and the numerical method. This issue is treated in Chapter 2. 1.3 Numerical Methods for Viscous Incompressible Flow Numerical algorithms for solving the incompressible Navier-Stokes system of equa- tions were first developed by Harlow and Welch [39] and Chorin [15, 16]. Descendants of these approaches are popular today. Harlow and Welch introduced the important contribution of the staggered-grid location of the dependent variables. On a stag- gered grid, the discrete Laplacian appearing in the derivation of the pressure-Poisson equation has the standard five-point stencil. On colocated grids it still has a five- point form but, if the central point is located at (i,j), the other points which are involved are located at (i+2,j), (i-2j), (i,j+2), and (ij-2). Without nearest-neighbor linkages, two uncoupled ("checkerboard") pressure fields can develop independently. This pressure-decoupling can cause stability problems, since nonphysical discontinu- ities in the pressure may develop [50]. In the present work, the velocity components are staggered one-half of a control volume to the west and south of the pressure which is defined at the center of the control volume as shown in Figure 1.1. Figure 1.1 also shows the locations of all boundary velocity components involved in the discretization and numerical solution, and representative boundary control volumes for u, v, and p. In Chorin's artificial compressibility approach [15] a time-derivative of pressure is added to the continuity equation. In this manner the continuity equation becomes an equation for the pressure, and all the equations can be integrated forward in time, either as a system or one at a time. The artificial compressibility method is closely related to the penalty formulation used in finite-element methods [41]. The equations are solved simultaneously in finite-element formulations. Penalty methods and the artificial compressibility approach suffer from ill-conditioning when the equations have strong nonlinearities or source terms. Because the pressure term is artificial, they are not time-accurate either. Projection methods [16, 62] are two-step procedures which first obtain a velocity field by integrating the momentum equations, and then project this vector field into a divergence-free space by subtracting the gradient of the pressure. The pressure- Poisson equation is solved to obtain the pressure. The solution must be obtained to a high degree of accuracy in unsteady calculations in order to obtain the correct long-term behavior [76]-every step may therefore be fairly expensive. Furthermore, the time-step size is limited by stability considerations, depending on the implicitness of the treatment used for the convection terms. "Pressure-based" methods for the incompressible Navier-Stokes equations include SIMPLE [61] and its variants, SIMPLEC [19], SIMPLER [60], and PISO [43]. These methods are similar to projection methods in the sense that a non-mass-conserving velocity field is computed first, and then corrected to satisfy continuity. However, they are not implicit in two steps because the nonlinear convection terms are linearized explicitly. Instead of a pressure-Poisson equation, an approximate equation for the pressure or pressure-correction is derived by manipulating the discrete forms of the momentum and continuity equations. A few iterations of a suitable relaxation method are used to obtain a partial solution to the system of correction equations, and then new guesses for pressure and velocity are obtained by adding the corrections to the old values. This process is iterated until all three equations are satisfied. The iterations require underrelaxation because of the sequential coupling between variables. Compared to projection methods, pressure-based methods are less implicit when used for time-dependent problems. However, they can be used to seek the steady-state directly if desired. Compared to a fully coupled strategy, the sequential pressure-based approach typically has slower convergence and less robustness with respect to Reynolds num- ber. However, the sequential approach has the important advantage that additional complexities, for example, chemical reaction, can be easily accommodated by simply adding species-balance equations to the stack. The overall run time increases since each governing equation is solved independently, and the total storage requirements scale linearly with the number of equations solved. On the other hand, the computer time and storage requirements escalate faster in a fully coupled solution strategy. The typical way around this problem is to solve simultaneously the continuity and momen- tum equations, then solve any additional equations in a sequential fashion. Without knowing beforehand that the pressure-velocity coupling is the strongest among all the various flow variables, however, the extra computational effort spent in simultaneous solution of these equations is unwarranted. There are other approaches for solving the incompressible Navier-Stokes equa- tions, notably methods based on vorticity-streamfunction (w-p) or velocity-vorticity (u-w) formulations, but pressure-based methods are easier, especially with regard to boundary conditions and possible extension to 3-d domains. Furthermore, they have demonstrated considerable robustness in computing incompressible flows. A broad range of applications of pressure-based methods is demonstrated in [73]. 1.4 Parallel Computing General background of parallel computers and their application to the numeri- cal solution of partial differential equations is given in Hockney and Jesshope [40] and Ortega and Voigt [58]. Fischer and Patera [23] gave a recent review of parallel computing from the perspective of the fluid dynamics community. Their "indirect cost," the parallel run time, is of primary interest here. The "direct cost" of parallel computers and their components is another matter entirely. For the iteration-based numerical methods developed here, the parallel run time is the cost per iteration multiplied by the number of iterations. The latter is affected by the characteristics of the particular parallel computer used and the algorithms and implementations em- ployed. Parallel computers come in all shapes and sizes, and it is becoming virtually impossible to give a thorough taxonomy. The background given here is limited to a description of the type of computer used in this work. 1.4.1 Data-Parallelism and SIMD Computers Single-instruction stream/multiple-data stream (SIMD) computers include the connection machines manufactured by the Thinking Machines Corporation, the CM and CM-2, and the MP-1, MP-2, and MP-3 computers produced by the MasPar Cor- poration. These are massively-parallel machines consisting of a front-end computer and many processor/memory pairs, figuratively, the "back-end." The back-end pro- cessors are connected to each other by a "data network." The topology of the data network is a major feature of distributed-memory parallel computers. The schematic in Figure 1.2 gives the general idea of the SIMD layout. The program executes on the serial front-end computer. The front-end triggers the syn- chronous execution of the "back-end" processors by sending "code blocks" simul- taneously to all processors. Actually, the code blocks are sent to an intermediate "control processorss)" The control processor broadcasts the instructions contained in the code block, one at a time, to the computing processors. These "front-end- to-processor" communications take time. This time is an overhead cost not present when the program runs on a serial computer. The operands of the instructions, the data, are distributed among the processors' memories. Each processor operates on its own locally-stored data. The "data" in grid-based numerical methods are the arrays, 2-d in this case, of dependent variables, geometric quantities, and equation coefficients. Because there are usually plenty of grid points and the same governing equations apply at each point, most CFD algorithms contain many operations to be performed at every grid point. Thus this "data-parallel" approach is very natural to most CFD algorithms. Many operations may be done independently on each grid point, but there is cou- pling between grid points in physically-derived problems. The data network enters the picture when an instruction involves another processor's data. Such "interpro- cessor" communication is another overhead cost of solving the problem on a parallel computer. For a given algorithm, the amount of interprocessor communication de- pends on the "data mapping," which refers to the partitioning of the arrays and the assignment of these "subgrids" to processors. For a given machine, the speed of the interprocessor communication depends on the pattern of communication (random or regular) and the distance between the processors (far away or nearest-neighbor). The run time of a parallel program depends first on the amount of front-end and parallel computation in the algorithm, and the speeds of the front-end and back- end for doing these computations. In the programs developed here, the front-end computations are mainly the program control statements (IF blocks, DO loops, etc.). The front-end work is not sped up by parallel processing. The parallel computations are the useful work, and by design one hopes to have enough parallel computation to amortize both the front-end computation and the interprocessor and front-end-to- processor communication, which are the other factors that contribute to the parallel run time. From this brief description it should be clear that SIMD computers have four char- acteristic speeds: the computation speed of the processors, the communication speed between processors, and the speed of the front-end-to-processor communication, i.e. the speed that code blocks are transferred, and the speed of the front-end. These machine characteristics are not under the control of the programmer. However, the amount of computation and communication a program contains is determined by the programmer because it depends on the algorithm selected and the algorithm's imple- mentation (the choice of the data mapping, for example). Thus, the key to obtaining good performance from SIMD computers is to pick a suitable algorithm, "matched" in a sense to the architecture, and to develop an implementation which minimizes and localizes the interprocessor communication. Then, if there is enough parallel computation to amortize the serial content of the program and the communication overheads, the speedup obtained will be nearly the number of processors. The actual performance, because it depends on the computer, the algorithm, and the imple- mentation, must be determined by numerical experiment on a program-by-program basis. SIMD computers are restricted to exploiting data-parallelism, as opposed to the parallelism of the tasks in an algorithm. The task-parallel approach is more com- monly used, for example, on the Cray C90 supercomputer. Multiple-instruction stream/multiple-data stream (MIMD) computers, on the other hand, are composed of more-or-less autonomous processor/memory pairs. Examples include the Intel series of machines (iPSC/2, iPSC/860, and Paragon), workstation clusters, and the connec- tion machine CM-5. However, in CFD, the data-parallel approach is the prevalent one even on MIMD computers. The front-end/back-end programming paradigm is implemented by selecting one processor to initiate programs on the other processors, accumulate global results, and enforce synchronization when necessary, a strategy called single-program-multiple-data (SPMD) [23]. The CM-5 has a special "control network" to provide automatic synchronization of the processor's execution, so a SIMD programming model can be supported as well as MIMD. SIMD is the manner in which the CM-5 has been used in the present work. The advantage to using the CM-5 in the SIMD mode is that the programmer does not have to explicitly specify message-passing. This simplification saves effort and increases the effective speed of communication because certain time-consuming protocols for the data transfer can be eliminated. 1.4.2 Algorithms and Performance The previous subsection discussed data-parallelism and SIMD computers, i.e. what parallel computing means in the present context and how it is carried out by SIMD-type computers. To develop programs for SIMD computers requires one to recognize that unlike serial computers, parallel computers are not black boxes. In addition to the selection of an algorithm with ample data-parallelism, consideration must be given to the implementation of the algorithm in specific ways in order to achieve the desired benefits speedupss over serial computations). The success of the choice of algorithm and the implementation on a particular computer is judged by the "speedup" (S) and "efficiency" (E) of the program. The communications mentioned above, front-end-to-processor and interprocessor, are es- sentially overhead costs associated with the SIMD computational model. They would not be present if the algorithm were implemented on a serial computer, or if such communications were infinitely fast. If the overhead cost was zero, a parallel program executing on n, processors would run np times faster than on a single processor, a speedup of np. This idealized case would also have a parallel efficiency of 1. The parallel efficiency E measures the actual speedup in comparison with the ideal. One is also interested in how speedup, efficiency, and the parallel run time (Tp) scale with problem size, and with the number of processors used. The objective in using parallel computers is more than just obtaining a good speedup on a particular problem size and a particular number of processors. For parallel CFD, the goals are to either (1) reduce the time (the indirect cost [23]) to solve problems of a given complexity, to satisfy the need for rapid turnaround times in design work, or (2) increase the complexity of problems which can be solved in a fixed amount of time. For the iteration-based numerical methods studied here, there are two considerations: the cost per iteration, and the number of iterations, respectively, computational and numerical factors. The total run time is the product of the two. Gustafson [35] has presented fixed-size and scaled-size experiments whose results describe how the cost per iteration scales on a particular machine. In the fixed- size experiment, the efficiency is measured for a fixed problem size as processors are added. The hope is that the run time is halved when the number of processors is doubled. However, the run time obviously cannot be reduced indefinitely by adding more processors because at some point the parallelism runs out-the limit to the attainable speedup is the number of grid points. In the scaled-size experiment, the problem size is increased along with the number of processors, to maintain a constant local problem size for each of the parallel processors. Care must be taken to make timings on a per iteration basis if the number of iterations to reach the end of the computation increases with the problem size. The hope in such an experiment is that the program will maintain a certain high level of parallel efficiency E. The ability to maintain E in the scaled-size experiment indicates that the additional processors increased the speedup in a one-for-one trade. 1.5 Pressure-Based Multigrid Methods Multigrid methods are a potential route to both computationally and numerically scalable programs. Their cost per iteration on parallel computers and convergence rate is the subject of Chapters 4-5. For sufficiently smooth elliptic problems, the convergence rate of multigrid methods is independent of the problem size-their op- eration count is O(N). In practice, good convergence rates are maintained as the problem size increases for Navier-Stokes problems, also, provided suitable multigrid components-the smoother, restriction and prolongation procedures-and multigrid techniques are employed. The standard V-cycle full-multigrid (FMG) algorithm has an almost optimal operation count, O(log2N) for Poisson equations, on parallel com- puters. Provided the multigrid algorithm is implemented efficiently and that the cost per iteration scales well with the problem size and the number of processors, the multigrid approach seems to be a promising way to exploit the increased computa- tional capabilities that parallel computers offer. The pressure-based methods mentioned previously involve the solution of three systems of linear algebraic equations, one each for the two velocity components and one for the pressure, by standard iterative methods such as successive line- underrelaxation (SLUR). Hence they inherit the convergence rate properties of these solvers, i.e. as the problem size grows the convergence rate deteriorates. With the single-grid techniques, therefore, it will be difficult to obtain reasonable turnaround times when the problem size is increased into the target range for parallel com- puters. Multigrid techniques for accelerating the convergence of pressure-correction methods should be pursued, and in fact they have been within the last five or so years [70, 74, 80]. However, there are still many unsettled issues. The complexities affecting the convergence rate of single-grid calculations carry over to the multigrid framework and are compounded there by the coupling between the evolving solutions on multiple grid levels, and by the particular "grid-scheduling" used. Linear multigrid methods have been applied to accelerate the convergence rate for the solution of the system of pressure or pressure-correction equations [4, 22, 42, 64, 94]. However, the overall convergence rate does not significantly improve because the velocity-pressure coupling is not addressed [4, 22]. Therefore the multigrid strategy should be applied on the "outer loop," with the role of the iterative relaxation method played by the numerical methods described above, e.g. the projection method or the pressure-correction method. Thus, the generic term "smoother" is prescribed because it reflects the purpose of the solution of the coupled system of equations going on inside the multigrid cycle-to smooth the residual so that an accurate coarse-grid approximation of the fine-grid problem is possible. It is not true that a good solver, one with a fast convergence rate on single-grid computations, is necessarily a good smoother of the residual. It is therefore of interest to assess pressure-correction meth- ods as potential multigrid smoothers. See Shyy and Sun [74] for more information on the staggered-grid implementation of multigrid methods, and some encouraging results. Staggered grids require special techniques [21, 74] for the transfer of solutions and residuals between grid levels, since the positions of the variables on different levels do not correspond. However, they alleviate the "checkerboard" pressure stability problem [50], and since techniques have already been established [74], there is no reason not to go this route, especially when cartesian grids are used as in the present work. Vanka [89] has proposed a new numerical method as a smoother for multigrid computations, one which has inferior convergence properties as a single-grid method but apparently yields an effective multigrid method. A staggered-grid finite-volume discretization is employed. In Vanka's smoother, the velocity components and pres- sure of each control volume are updated simultaneously, so it is a coupled approach, but the coupling between control volumes is not taken into account, so the calcu- lation of new velocities and pressures is explicit. This method is sometimes called the "locally-coupled explicit" or "block-explicit" pressure-based method. The control volumes are visited in lexicographic order in the original method which is therefore aptly called BGS (block Gauss-Seidel). Line-variants have been developed to couple the flow variables in neighboring control volumes along lines (see [80, 87]). Linden et al.[50] gave a brief survey of multigrid methods for the steady-state in- compressible Navier-Stokes equations. They argue without analysis that BGS should be preferred over the pressure-correction type methods since the strong local cou- pling is likely to have better success smoothing the residual locally. On the other hand, Sivaloganathan and Shaw [71, 70] have found good smoothing properties for the pressure-correction approach, although the analysis was simplified considerably. Sockol [80] has compared the point and line-variants of BGS with the pressure- correction methods on serial computers, using model problems with different physical characteristics. SIMPLE and BGS emerge as favorites in terms of robustness with BGS preferred due to a lower cost per iteration. This preference may or may not carry over to SIMD parallel computers (see Chapter 4 for comparison). Interesting applications of multigrid methods to incompressible Navier-Stokes flow problems can be found in [12, 28, 48, 54]. In terms of parallel implementations there are far fewer results although this field is rapidly growing. Simon [77] gives a recent cross-section of parallel CFD results. Parallel multigrid methods, not only in CFD but as a general technique for partial differential equations, have received much attention due to their desirable O(N) operation count on Poisson equations. However, it is apparently difficult to find or design parallel computers with ideal communication networks for multigrid [13]. Consequently implementations have been pursued on a variety of machines to see what performance can be obtained with the present generation of parallel machines, and to identify and understand the basic issues. Dendy et al.[18] have recently described a multigrid method on the CM-2. However, to accommodate the data- parallel programming model they had to dimension their array data on every grid level to the dimension extents of the finest grid array data. This approach is very wasteful of storage. Consequently the size of problems which can be solved is greatly reduced. Recently an improved release of the compiler has enabled the storage problem to be circumvented with some programming diligence (see Chapter 5). The implementation developed in this work is one of the first to take advantage of the new compiler feature. In addition to parallel implementations of serial multigrid algorithms, several novel multigrid methods have been proposed for SIMD computers [25, 26, 33]. Some of the algorithms are instrinsically parallel [25, 26] or have increased parallelism because they use multiple coarse grids, for example [33]. These efforts and others have been recently reviewed [14, 53, 92]. Most of the new ideas have not been developed yet for solving the incompressible Navier-Stokes equations. One of the most prominent concerns addressed in the literature regarding parallel implementations of serial multigrid methods is the coarse grids. When the number of grid points is smaller than the number of processors the parallelism is reduced to the number of grid points. This loss of parallelism may significantly affect the parallel efficiency. One of the routes around the problem is to use multiple coarse grids [59, 33, 79]. Another is to alter the grid-scheduling to avoid coarse grids. This approach can lead to computationally scalable implementations [34, 49] but may sacrifice the convergence rate. "Agglomeration" is an efficiency-increasing technique used in MIMD multigrid programs which refers to the technique of duplicating the coarse grid problem in each processor so that computation proceeds independently (and redundantly). Such an approach can also be scalable [51]. However, most atten- tion so far has focused on parallel implementations of serial multigrid algorithms, in particular on assessing the importance of the coarse-grid smoothing problem for dif- ferent machines and on developing techniques to minimize the impact on the parallel efficiency. 1.6 Description of the Research The dissertation is organized as follows. Chapter 2 discusses the role of the mass conservation in the numerical consistency of the single-grid SIMPLE method for open boundary problems, and explains the relevance of this issue to the convergence rate. In Chapter 3 the single-grid pressure-correction method is implemented on the MP-1, CM-2, and CM-5 computers and its performance is analyzed. High parallel efficien- cies are obtained at speeds and problem sizes well beyond the current performance of such algorithms on traditional vector supercomputers. Chapter 4 develops a multigrid numerical method for the purpose of accelerating the single-grid pressure-correction method and maintaining the accelerated convergence property independent of the problem size. The multigrid smoother, the intergrid transfer operators, and the sta- bilization strategy for Navier-Stokes computations are discussed. Chapter 5 describes the actual implementation of the multigrid algorithm on the CM-5, its convergence rate, and its parallel run time and scalability. The convergence rate depends on the 18 flow problem and the coarse-grid discretization, among other factors. These factors are considered in the context of the "full-multigrid" (FMG) starting procedure by which the initial guess on the fine grid is obtained. The cost of the FMG proce- dure is a concern for parallel computation [88], and this issue is also addressed. The results indicate that the FMG procedure may influence the asymptotic convergence rate and the stability of the multigrid iterations. Concluding remarks in each chapter summarize the progress made and suggest avenues for further study. Figure 1.1. Staggered-grid layout of dependent variables, for a small but complete domain. Boundary values involved in the computation are shown. Representative u, v, and pressure boundary control volumes are shaded. short blocks of parallel code Sequencer (CM-2) Array control unit (MP-1) Multiple SPARC nodes (CM-5: individual instruction P.E. P. E. P. E. P. E. more P.E.s 0 0 0 array data partitioned among processor memories Interprocessor communication network hypercube (CM-2) + "NEWS" 3-stage crossbar (MP-1) + "X-Net" fat tree (CM-5) Figure 1.2. Layout of the MP-1, CM-2, and CM-5 SIMD computers. Front End (CM-2 and MP-1) Partition Manager (CM-5) -> serial code, control code, scalar data a 0 a * 0 0 CHAPTER 2 PRESSURE-CORRECTION METHODS 2.1 Finite-Volume Discretization on Staggered Grids The formulation of the numerical method used in this work begins with the inte- gration of the governing equations Eq 1.1-1.3 over each of the control volumes in the computational domain. Figure 1.1 shows a model computational domain with u, v, and p (cell-centered) control volumes shaded. The continuity equation is integrated over the p control volumes. Consider the discretization of the u-momentum equation for the control volume shown in Figure 2.1 whose dimensions are Ax and Ay. The v control volumes are done exactly the same except rotated 900. Integration of Eq. 1.2 over the shaded region is interpreted as follows for each of the terms: I Opu Opup P z dxdy AAy, (2.1) at at 2 dx dy = (pu pu ) Ay, (2.2) audx dy = (puv, pu,sv) A, (2.3) S- dx dy = -(p p,)Ay, (2.4) I (" ( au It2U dxdy = a Ax (2.6) jXdx = d2e-i jA (2.5) / P a dady=Pau Itau A, X (2.6) if 49y 2 9ay ys) The lowercase subscripts e, w. n, s indicate evaluation on the control volume faces. By convention and the mean-value theorem, these are at the midpoint of the faces. The subscript P in Eq. 2.1 indicates evaluation at the center of the control volume. 21 Because of the staggered grid, the required pressure values in Eq. 2.4 are already located on the u control volume faces. The pressure-gradient term is effectively a second-order central-difference approximation. With colocated grids, however, the control-volume face pressures are obtained by averaging the nearby pressures. This averaging results in the pressure at the cell center dropping out of the expression for the pressure gradient. The central-difference in Eq. 2.4 is effectively taken over a distance 2Ax on colocated grids. Thus staggered cartesian grids provide a more accurate approximation of the pressure-gradient term since the difference stencil is smaller. The next step is to approximate the terms which involve values at the control volume faces. In Eq. 2.2, one of the ue and one of the u, are replaced by an average of neighboring values, 2 2) A ( UE+ Up UP+ UW ) (ue pu y = p 2 e P U2 A (2.7) and in Eq. 2.3, v, and v, are obtained by averaging nearby values, ( Vne + nw Vse + Vs (pUnVn pusv) Ax= p V -u pV us), Ax (2.8) The remaining face velocities in the convection terms, u,, u,, ue, and uw, are ex- pressed as a certain combination of the nearby u values-which u values are involved and what weighting they receive is prescribed by the convection scheme. Some pop- ular recirculating flow convection schemes are described in [73, 75]. The control-volume face derivatives in the diffusion terms are evaluated by central differences, P Ou 9u E-UP P-UW A (2.9) P -I a AX = M UN- P p Ax (2.10) dy ay Ay Ay The unsteady term in Eq. 2.1 is approximated by a backward Euler scheme. All the terms are evaluated at the "new" time level, i.e. implicitly. Thus, the discretized momentum equations for each control volume can be put into the following general form, apup = aEUE + awuw + aNUN + asus + b, (2.11) where b = (pw -pe)Ay+pun/At, the superscript n indicating the previous time-step. The coefficients aN, as, etc. are comprised of the terms which modify UN, us, etc. in the discretized convection and diffusion terms. The continuity equation is integrated over a pressure control volume, I/ [ pu+ dx dy = p(ue u)Ay + p(vn v)A = 0. (2.12) Again the staggered grid is an advantage because the normal velocity components on each control volume face are already in position-there is no need for interpolation. 2.2 The SIMPLE Method One SIMPLE iteration takes initial velocity and pressure fields (u*, v*,p*) and computes new guesses (u, v,p). The intermediate values are denoted with a tilde, (it, j,). In the algorithm below, au(u*, v*), for example, means that the aN coeffi- cient in the u-momentum equation depends on u* and v*. The parameters v,, v,, and vc are the numbers of "inner" iterations to be taken for the u, v, and continuity equa- tions, respectively. This notation will be clarified by the following discussion. The inner iteration count is indicated by the superscript enclosed in parentheses. Finally, wU and w, are the relaxation factors for the momentum and continuity equations. SIMPLE (u*, v*, p*; Vu, vV, Vp, WU, wc) Compute u coefficients au(u*, v*) (k = P,E,W,N,S) and source term b"(u*,p*) for each discrete u-momentum equation: a~Up = aNiUN + auis + auiE + a'viw + bU + (1 wu -2-u UJUV s Ev W JUV P Do v, iterations to obtain an approximate solution for ii starting with u* as the initial guess u(n) = Gu(n-1) + fU f = un=u) Compute v coefficients ak((i, v*) (k = E,W,N,S) and source term bv(v*,p*) for each discrete v-momentum equation: v2-p = a'VNl + a'ls + as + IE + a'w w + b + (1 wuv,) v Do v iterations to obtain an approximate solution for v starting with v* as the initial guess v(n) = Gv(n-l1) + fV V = v(n=v) Compute p' coefficients a' (k = P,E,W,N,S) and source term bc(ii, 3) for each discrete p' equation: app p = aNp N + asp s + aEP'E + awP'w + bc Do vc iterations to obtain an approximate solution for p' starting with zero as the initial guess p'(") = Gp'("-1) + f Correct f, i, and p* at every interior grid point up = ip + , (ap')p Vp = pp + (a')p pp = p* + WcP'p The algorithm is not as complicated as it looks. The important point to note is that the major tasks to be done are the computing of coefficients and the solving of the systems of equations. The symbol G indicates the iteration matrix of whatever type relaxation is used on these inner iterations (SLUR in this case), and f is the corresponding source term. In the SIMPLE pressure-correction method [61], the averages in Eq. 2.7 and 2.8 are lagged in order to linearize the resulting algebraic equations. The governing equations are solved sequentially. First, the u momentum equation coefficients are computed and an updated u field is computed by solving the system of linear alge- braic equations. The pressures in Eq. 2.4 are lagged. The v momentum equation is solved next to update v. The continuity equation, recast in terms of pressure correc- tions, is then set up and solved. These pressure corrections are coupled to velocity corrections. Together they are designed to correct the velocity field so that it satisfies the continuity constraint, while simultaneously correcting the pressure field so that momentum conservation is maintained. The relationship between the velocity and pressure corrections is derived from the momentum equation, as described in the next section. The resulting system of equations is fully coupled, as one might expect knowing the elliptic nature of pressure in incompressible fluids, and is therefore expensive to solve. However, if the resulting system of pressure-correction equations were solved exactly, the divergence- free constraint and the momentum equations (with old values of u and v present in the nonlinear convection terms) would be satisfied. This approach would constitute an implicit method of time integration for the linearized equations. The time-step size would have to be limited to avoid stability problems caused by the linearization. To reduce the computational cost, the SIMPLE prescription is to use an approx- imate relationship between the velocity and pressure corrections (hence the label "semi-implicit"). Variations on the original SIMPLE approximation have shown bet- ter convergence rates for simple flow problems, but in discretizations on curvilinear grids and other problems with significant contributions from source terms, the per- formance is no better than the original SIMPLE method (see the results in [4]). The goal of satisfying the divergence-free constraint can still be attained, if the system of pressure-correction equations is converged to strict tolerances, because the discrete continuity equations are still being solved. But satisfaction of the momentum equations cannot be maintained with the approximate relationship. Consequently it is no longer desirable to solve the p'-system of equations to strict tolerances. It- erations are necessary to find the right velocities and pressures which satisfy all three equations. Furthermore, since the equation coefficients are changing from one iteration to the next, it is pointless to solve the momentum equations to strict tol- erances. In practice, only a few iterations of a standard scheme such as successive line-underrelaxation (SLUR) are performed. The single "outer" iteration outlined above is repeated many times, with under- relaxation to prevent the iterations from diverging. In this sense a two-level iterative procedure is being employed. In the outer iterations, the momentum and pressure- correction equations are iteratively updated based on the linearized coefficients and sources, and inner iterations are applied to partially solve the systems of linear alge- braic equations. The fact that only a few inner iterations are taken on each system of equations sug- gests that the asymptotic convergence rate of the iterative solver, which is the usual means of comparison between solvers, does not necessarily dictate the convergence rate of the outer iterative process. Braaten and Shyy [4] have found that the con- vergence rate of the outer iterations actually decreases when the pressure-correction equation is solved to a much stricter tolerance than the momentum equations. They concluded that the balance between the equations is important. Because u, v, and p' are segregated, the overall convergence rate is strongly dependent on the partic- ular flow problem, the grid distribution and quality, and the choice of relaxation parameters. In contrast to projection methods, which are two-step but treat the convection terms explicitly (or more recently by solving a Riemann problem [2]) and are therefore restricted from taking too large a time-step, the pressure-correction approach is fully implicit with no time-step limitation, but many iterations may be necessary. The projection methods are formalized as time-integration techniques for semi-discrete equations. SIMPLE is an iterative method for solving the discretized Navier-Stokes system of coupled nonlinear algebraic equations. But the details given above should make it clear that these techniques bear strong similarities-specifically, a single SIMPLE iteration would be a projection method if the system of pressure-correction equations was solved to strict tolerances at each iteration. It would be interesting to do some numerical comparisons between projection methods and pressure-correction methods to further clarify the similarity. 2.3 Discrete Formulation of the Pressure-Correction Equation The discrete pressure-correction equation is obtained from the discrete momentum and continuity equations as follows. The velocity field which has been newly obtained by solving the momentum equations was denoted by (ii, v) earlier. The pressure field after the momentum equations are solved still has the initial value p*. So fi, u, and p* satisfy the u-momentum equation apUip = aEUE + awuw + aNUN + asfis + (p* p*)Ay, (2.13) and the corresponding v-momentum equation. The corrected (continuity-satisfying) velocity field (u, v) satisfies the u-momentum equation with the corrected pressure field p, apup = aEUE + awuw + aNUN + asus + (pw pe)Ay, (2.14) and likewise for the v-momentum equation. Additive corrections are assumed, i.e. u = ii + u' (2.15) v = v' (2.16) p= p* + p'. (2.17) Subtracting Eq. 2.13 from Eq. 2.14 gives the desired relationship between pressure and the u corrections, apup = akk + (p, p')Ay, (2.18) k=E,W,N,S with a similar expression for the v corrections. If Eq. 2.18 is used as is, then the nearby velocity corrections in the summation need to be replaced by similar expressions involving pressure-corrections. This requirement brings in more velocity corrections and more pressure corrections, and so on, leading to an equation which involves the pressure corrections at every grid point. The resulting system of equations would be expensive to solve. Thus, the summation term is dropped in order to obtain a compact expression for the velocity correction in terms of pressure corrections. At convergence, the pressure corrections (and therefore the velocity corrections) go to zero, so the precise form of the approximate pressure- velocity correction relationship does not figure in the final converged solution. The discrete form of the pressure-correction equation follows by first substituting the simplified version of Eq. 2.18 into Eq. 2.15, Up = Up + Up = Up + (p, p')Ay, (2.19) and then substituting this into the continuity equation Eq. 2.12, (with an analogous formula for vp). The result is pAy2 pAy2 PAx2 I PAX2 a (pu),-P) -(P'' (P-PP)+ (PP -pN) -- (-P') = b, (2.20) ap(ue) ap(uw) ap(vn) ap(v,) where the source term b is b = p,Ay piiAy + pv;:A pvAx (2.21) Recall that Eq. 2.20 and Eq. 2.21 are written for the pressure control volumes, so that there is some interpretation required. The term ap(ue) in Eq. 2.20 is the appropriate ap for the discretized u-momentum equation, Eq. 2.13. In other words, up in Eq. 2.13 is actually u,, u,, u,, or us in Eq. 2.20 and 2.21, relative to the pressure control volumes on the staggered grid. Eq. 2.20 can be rearranged into the same general form as Eq. 2.11. From Eq. 2.21, it is apparent that the right-hand side term is the net mass flux entering the control volume, which should be zero in incompressible flow. In the formulation of the pressure-correction equation for boundary control vol- umes, one makes use of the fact that the normal velocity components on the bound- aries are known from either Dirichlet or Neumann boundary conditions, so no velocity correction is required there. Consequently, the formulation of Eq. 2.20 for boundary control volumes does not require any prescription of boundary p' values [60] when velocity boundary conditions are prescribed. Without the summation from Eq. 2.18, it is apparent that a zero velocity correction for the outflow boundary u-velocity component is obtained when pw = pe-in effect, a Neumann boundary condition on pressure is implied. This boundary condition is appropriate for an incompressible fluid because it is physically consistent with the governing equations in which only the pressure gradient appears. There is a unique pressure gradient but the level is adjustable by any constant amount. If it happens that there is a pressure specified on the boundary, for example by Eq. 1.4, then the correction there will be zero, pro- viding a boundary condition for Eq. 2.20. Thus, it seems that there are no concerns over the specification of boundary conditions for the p' equations. 2.4 Well-Posedness of the Pressure-Correction Equation 2.4.1 Analysis To better understand the characteristics of the pressure-correction step in the SIMPLE procedure, consider a model 3 x 3 computational domain, so that 9 algebraic equations for the pressure corrections are obtained. Number the control volumes as shown in Figure 2.3. Then the system of p' equations can be written al -a 0 -ak 0 0 0 0 0 p' p(u\ u1 + v1 v,) 2 2 -aw a --a 0 -a 0 0 0 0 p' p(u + v v ) 0 -a, a 0 0 -aN 0 0 0 p' P(u U + -v) -a 0 0 a -4 0-aa4 0 0 0 4 p(u -u+ v -v) 55 + 5 _V5) S-4 0 -a, a -aE 0 -aN 0 p' = p(uW-n+v -v) (2.22) 0 0 -a 0 -a6 ap 0 0 -aN P'6 P(u6 + v v,) 0 0 0 -as 0 0 ap -aE 0 p'7 p(u7 u7 + v v7) S 0 0 -a -a8 as -4a P' P(u8 u+ v8 - S0 0 0 -a 0 -a9 a9 .Pg p( u + v9 v9) where the superscript designates the cell location and the subscript designates the coefficient linking the point in question, P, and the neighboring node. The right-hand side velocities are understood to be tilde quantities as in Eq. 2.21. In finite-volume discretizations, fluxes are estimated at the control volume faces which are common to adjacent control volumes, so if the governing equations are cast in conservation law form, as they are here, the discrete efflux of any quantity out of one control volume is guaranteed to be identical to the influx into its neighbor. There is no possibility of internal sources or sinks. In fact this is what makes finite- volume discretizations preferable to finite-difference discretizations. The following relationships, using control volume 5 in Figure 2.3 as an example, follow from Eq. 2.20 and the internal consistency of finite-volume discretizations: a = a + as + a N + aG (2.23) a = a E = aw, aN =as as = aN (2.24) 5 = = 4 =1 v, = 2 (2.25) w e e zw n S Vn Eq. 2.23 states that the coefficient matrix is pentadiagonal and diagonally dominant for the interior control volumes. Furthermore, when the natural boundary condition (zero velocity correction) is applied, the appropriate term in Eq. 2.20 for the boundary under consideration does not appear, and therefore the pressure-correction equations for the boundary control volumes also satisfy Eq. 2.23. If a pressure boundary condi- tion is applied so that the corresponding pressure correction is zero, then one would set p' = 0 in Eq. 2.20, for example, which would give aw + aN + as < ap. Thus, either way, the entire coefficient matrix in Eq. 2.22 is diagonally dominant. However, with the natural prescription for boundary treatment, no diagonal term exceeds the sum of its off-diagonal terms. Thus, the system of equations Eq. 2.22 is linearly dependent with the natural (velocity) boundary conditions, which can be verified by adding the 9 equations above. Because of Eq. 2.23 and Eq. 2.24 all terms on the left-hand side of Eq. 2.22 identically cancel one another. At all interior control volume interfaces, the right- hand side terms identically cancel due to Eq. 2.25, and the remaining source terms are simply the boundary mass fluxes. This cancellation is equivalent to a discrete statement of the divergence theorem SV-idQ = j i -d(0fl) (2.26) where 0f is the domain under consideration and n is the unit vector in the direction normal to its boundary 0t. Due to the linear dependence of the left-hand side of Eq. 2.22, the boundary mass fluxes must also sum to zero in order for the system of equations to be consistent. No solution exists if the linearly dependent system of equations is inconsistent. The situation can be likened to a steady-state heat conduction problem with source terms and adiabatic boundaries. Clearly, a steady-state solution only exists if the sum of the source terms is zero. If there is a net heat source, then the temperature inside the domain will simply rise without bound if an iterative solution strategy (quasi time-marching) is used. Likewise, the net mass source in flow problems with open boundaries must sum to zero for the pressure-correction equation to have a solution. In other words, global mass conservation is required in discrete form in order for a solution to exist. The interesting point to note is that during the course of SIMPLE iterations, when the pressure-correction equation is executed, the velocity field does not usually conserve mass globally in flow problems with open boundaries, unless explicit measure is taken to enforce global mass conservation. The purpose of solving the pressure-correction equations is to drive the local mass sources to zero by suitable velocity corrections. But the pressure-correction equations which are supposed to accomplish this purpose do not have a solution unless the net mass source is already zero. For domains with closed boundaries, global mass conservation is obviously not an issue. Furthermore, this problem does not only show up when the initial guess is bad. In the backward-facing step flow discussed below, the initial guess is zero everywhere except for inflow, which obviously is the worst case as far as a net mass source is concerned (all inflow and no outflow). But even if one starts with a mass-conserving initial guess, during the course of iterations the outflow velocity boundary condition which is necessary to solve the momentum equations will reset the outflow so that the global mass-conservation constraint is violated. 2.4.2 Verification by Numerical Experiments Support for the preceding discussion is provided by numerical simulation of two model problems, a lid-driven cavity flow and a backward-facing step flow. The con- figurations are shown along with other relevant data in Figure 2.2. Figure 2.4 shows the outer-loop convergence paths for the lid-driven cavity flow and the backward-facing step flow, both at Re = 100. The quantities plotted in Figure 2.4 are the logo of the global residuals for each governing equation obtained by summing up the local residuals, each of which is obtained by subtracting the left-hand side of the discretized equations from the right-hand side. For the cavity flow there are no mass fluxes across the boundary so, as mentioned earlier, the global mass conservation condition is always satisfied when the algorithm reaches the point of solving the system of p'-equations. The residuals have dropped to 107 after 150 iterations, which is very rapid convergence, indicating that good pressure and velocity corrections are being obtained. In the backward-facing step flow, however, the flowfield is very slow to develop because no global mass conservation measure is enforced. During the course of iter- ations, the mass flux into the domain from the left is not matched by an equal flux through the outflow boundary, and consequently the system of pressure-correction equations which is supposed to produce a continuity-satisfying velocity field does not have a solution. Correspondingly one observes that the outer-loop convergence rate is about 10 times worse than for cavity flow. Also, note that the momentum convergence path of the backward-facing step flow in Figure 2.4 tends to follow the continuity equation, indicating that the pressure and velocity fields are strongly coupled. The present flow problem bears some similarity to a fully-developed channel flow, in which the streamwise pressure-gradient and cross- stream viscous diffusion are balanced, so the observation that pressure and velocity are strongly coupled is intuitively correct. Thus, the convergence path is controlled by the development of the pressure field. The slow convergence rate problem is due to the inconsistency of the system of pressure-correction equations. The inner-loop convergence path (the SLUR iterations) for the p'-system of equa- tions must be examined to determine the manner in which the inner-loop inconsis- tency leads to poor outer-loop convergence rates. Table 2.1 shows leading eigenvalues for successive line-underrelaxation iteration matrices of the p'-system of equations at an intermediate iteration for which the outer-loop residuals had dropped to approx- imately 10-2. Largest 3 eigenvalues Cavity Flow Back-Step Flow A1 1.0 1.0 A2 0.956 0.996 A3 0.951 0.984 Table 2.1. Largest eigenvalues of iteration matrices during an intermediate itera- tion, applying the successive line-underrelaxation iteration scheme to the p'-system of equations. In both model problems the spectral radius is 1.0 because the p'-system of equa- tions is linearly dependent. The next largest eigenvalue is smaller in the cavity flow computation than in the step flow computation, which means a faster asymptotic con- vergence rate. However, the difference between 0.996 and 0.956 is not large enough to produce the significant difference observed in the outer convergence path. Figure 2.5 shows the inner-loop residuals of the SLUR procedure during an inter- mediate iteration. The two momentum equations are well-conditioned and converge to a solution within 4 iterations. In Figure 2.5 for the cavity flow case, the p'-equation converges to zero, although this happens at a slower rate than the two momentum equations because of the diffusive nature of the equation. In Figure 2.5 for the back- step flow, the inner-loop residual is fixed on a nonzero residual, which is in fact the initial level of inconsistency in the system of equations, i.e. the global mass deficit. Given that the system of p'- equations which is being solved does not satisfy the global continuity constraint, however, the significance or utility of the p'-field that has been obtained is unknown. In practice, the overall procedure may still be able to lead to a converged solu- tion, as in the present case. It appears that the outflow extrapolating procedure, a zero-gradient treatment utilized here, can help induce the overall computation to converge to the right solution [72]. Obviously, such a lack of satisfaction of global mass conservation is not desirable in view of the slow convergence rate. Further study suggests that the iterative solution to the inconsistent system of p'-equations converges on a unique pressure gradient, i.e. the difference between p' values at any two points tends to a constant value, even though the p'-field does not in general satisfy any of the equations in the system. This relationship is shown in Figure 2.6, in which the convergence of the difference in p' between the lower-left and upper-right locations in the domain of the cavity and backward-facing step flows is plotted. Also shown is the value of p' at the lower-left corner of the domain. For the cavity flow, there is a solution to the system of p'-equations, and it is obtained by the SLUR technique in about 10 iterations. Thus all the pressure corrections and the differences between them tend towards constant values. In the backward-facing step flow, however, the individual pressure corrections increase linearly with the number of iterations, symptomatic of the inconsistency in the system of equations. The differences between p' values approach a constant, however. The rate at which this unique pressure-gradient field is obtained depends on the eigenvalues of the iteration matrix. To resolve the inconsistency problem in the p'-system of equations and thereby improve the outer-loop convergence rate in the backward-facing step flow, global mass conservation has been explicitly enforced during the sequential solution procedure. The procedure used is to compute the global mass deficit and then add a constant value to the outflow boundary u-velocities to restore global mass conservation. Al- ternatively, corrections can be applied at every streamwise location by considering control volumes whose boundaries are the inflow plane, the top and bottom walls of the channel, and the i=constant line at the specified streamwise location. The artificially-imposed convection has the effect of speeding up the development of the pressure field, whose normal development is diffusion-dominated. It is interesting to note that this physically-motivated approach is in essence an acceleration of conver- gence of the line-iterative method via the technique called additive correction [45, 69]. The strategy is to adjust the residual on the current line to zero by adding a con- stant to all the unknowns in the line. This procedure is done for every line, for every iteration, and generally produces improvement in the SLUR solution of a system of equations. Kelkar and Patankar [45] have gone one step further by applying additive corrections like an injection step of a multigrid scheme, a so-called block correction technique. This technique is exploited to its fullest by Hutchinson and Raithby [42]. Given a fine-grid solution and a coarse grid, discretized equations for the correction quantities on the coarse grid are obtained by summing the equations for each of the fine-grid cells within a given coarse grid cell. A solution is then obtained (by direct methods in [45]) which satisfies conservation of mass and momentum. The corrections are then distributed uniformly to the fine grid cells which make up the coarse grid cell, and the iterative solution on the fine grid is resumed. However, experiences have shown that the net effect of such a treatment for complex flow problems is limited. Figure 2.7 illustrates the improved convergence rate of the continuity equation for the inner and outer loops, in the backward-facing step flow, when conservation of mass is explicitly enforced. The inner-loop data is from the 10th outer-loop iteration. In Figure 2.7, the cavity flow convergence path is also shown to facilitate the comparison. For the back-step, the overall convergence rate is improved by an order of magnitude, becoming slightly faster than the cavity flow case. This result reflects the improved inner-loop performance, also shown in Figure 2.7. The improved performance for the pressure-correction equation comes at the expense of a slightly slower convergence rate for the momentum equations, because of the nonlinear convection term. In short, it has been shown that a consistency condition, which is physically the re- quirement of global mass conservation, is critical for meaningful pressure-corrections to be guaranteed. Given natural (velocity) boundary conditions, which lead to a linearly dependent system of pressure-correction equations, satisfaction of the global continuity constraint is the only way that a solution can exist, and therefore the only way that the inner-loop residuals can be driven to zero. For the model backward- facing step flow in a channel with length L = 4 and a 21 x 9 mesh, the mass- conservation constraint is enforced globally or at every streamwise location by an additive-correction technique. This technique produces a 10-fold increase in the con- vergence rate. Physically, modifying the u velocities has the same effect as adding a convection term to the Poisson equation for the p'-field, which otherwise develops very slowly. A coarse grid size was used to demonstrate the need of enforcing global mass conservation. On a finer grid, this issue becomes more critical. In the next section, the solution accuracy aspects related to mass conservation will be addressed, and the computations will be conducted with more adequate grid resolution. 2.5 Numerical Treatment of Outflow Boundaries Continuing with the theme of well-posedness, the next numerical issue to be dis- cussed is the choice of outflow boundary location. If fluid flows into the domain at a boundary where extrapolation is applied, then, traditionally, the problem is not considered to be well-posed, because the information which is being transported into the domain does not participate in the solution to the problem [60]. Numerically, however, accurate solutions can be obtained using first-order extrapolation for the ve- locity components on a boundary where inflow is occurring [72]. Here open boundary treatment for both steady and time-dependent flow problems is investigated further. Figure 2.9 and 2.8 present streamfunction contours for a time-dependent flow problem, impulsively started backward-facing step flow, using central-differencing for the convection terms and first-order backward-differencing in time. A parabolic inflow velocity profile is specified, while outflow boundary velocities are obtained by first-order extrapolation. The Reynolds number based on the average inflow velocity Uavg and the channel height H, is 800. The expansion ratio H/h is 2 as in the model problem described in Figure 2.3. Time-accurate simulations were performed for two channel configurations, one with length L = 8 (81 x 41 mesh) and the other with length L = 16 (161 x 41 mesh). This flow problem has been the subject of some recent investigations focusing on open boundary conditions [30, 31]. For each time step, the SIMPLE algorithm is used to iteratively converge on a solution to the unsteady form of the governing equations, explicitly enforcing global conservation of mass during the course of iterations. In the present study, convergence was declared for a given time step when the global residuals had been reduced below 10-4. The time-step size was twice the viscous time scale in the y-direction, i.e. At = 2Ay2/v. Thus a fluid particle entering the domain at the average velocity u = 1 travels 2 units downstream during a time-step. Figure 2.8 shows the formation of alternate bottom/top wall recirculation regions during startup which gradually become thinner and elongated as they drift down- stream. For the L = 16 simulation (Figure 2.8), the transient flowfield has as many as four separation bubbles at T = 32, the latter two of which are eventually washed out of the domain. In the L = 8 simulation (Figure 2.9) the streamfunction plots are at times corresponding to those shown in Figure 2.8. Note that between T = 11 and T = 32, a secondary bottom wall recirculation zone forms and drifts downstream, exiting without reflection through the downstream boundary. The time evolution of the flowfield for the L = 8 and L = 16 simulations is virtually identical. As can be observed, the facts that a shorter channel length was used in Figure 2.9 and that a recirculating cell may go through the open boundary do not affect the solutions. Figure 2.10 compares the computed time histories of the bottom wall reattachment and top wall separation points between the two computations. The L = 8 and L = 16 curves are perfectly overlapped. The steady-state solutions for both the L = 8 and L = 16 channel configurations are also shown in Figure 2.9 and 2.8, respectively. Although the outflow boundary cuts the top wall separation bubble approximately in half, there is no apparent difference between the computed streamfunction contours for 0 < x < 8. Furthermore, the convergence rate is not affected by the choice of outflow boundary location. Figure 2.11 compares the steady-state u and v velocity profiles at x = 7 be- tween the two computations. The accuracy of the computed results is assessed by comparison with an FEM numerical solution reported by Gartling [27]. Figure 2.11 establishes quantitatively that the two simulations differ negligibly over 0 < x < 8 (the v profile differs on the order of 10-3) The velocity scale for the problem is 1. Neither v profile agrees perfectly with the solution obtained by Gartling, which may be attributed to the need for conducting further grid refinement studies in the present work and/or Gartling's work. Evidently the location of the open boundary is not critical to obtaining a con- verged solution. This observation indicates that the downstream information is com- pletely accounted for by the continuity equation. The correct pressure field can de- velop because the system of p'-equations requires only the boundary mass flux specifi- cation. If the global continuity constraint is satisfied, the pressure-correction equation is consistent regardless of whether there is inflow or outflow at the boundary where extrapolation is applied. The numerical well-posedness of the open boundary com- putation results in virtually identical flowfield development for the time-dependent L = 8 and L = 16 simulations as well as steady-state solutions which agree with each other and follow closely Gartling's benchmark data [27]. 2.6 Concluding Remarks In order for the SIMPLE pressure-correction method to be a well-posed numer- ical procedure for open boundary problems, explicit steps must be taken to ensure the numerical consistency of the pressure-correction system of equations during the course of iterations. For the discrete problem with the natural boundary treatment for pressure, i.e. normal velocity specified at all boundaries, global mass conserva- tion is the solvability constraint which must be satisfied in order that the system of p'-equations is consistent. Without a globally mass-conserving procedure enforced during each iterative step, the utility of the pressure-corrections obtained at each it- eration cannot be guaranteed. Overall convergence may still occur, albeit very slowly. In this regard, the poor outer-loop convergence behavior simply reflects the (poor) convergence rate of the inner-loop iterations of the SLUR technique. In general, the inner-loop residual is fixed on the value of the initial level of inconsistency of the system of p'-equations which physically is the global mass deficit. The convergence rate can be improved dramatically by explicitly enforcing mass conservation using an additive-correction technique. The results of numerical simulations of backward- facing step flow illustrate and support these conclusions. The mass-conservation constraint also has implications for the issue of proper numerical treatment of open boundaries where inflow is occurring. Specifically, the conventional viewpoint that inflow cannot occur at open boundaries without Dirich- let prescription of the inflow variables can be rebutted, based on the grounds that the numerical problem is well-posed if the normal velocity components satisfy the continuity constraint. Figure 2.1. Staggered grid u control volume and the nearby variables which are involved in the discretization of the u-momentum equation. U=1 ^t------ U(y) \ 1i\~ L 1 W> -------------- ^:^ __ 7 7/ h Figure 2.2. Description of two model problems. Both are at Re = 100. The cavity is a square with a top wall sliding to the left, while the backward-facing step is a 4 x 1 rectangular domain with an expansion ratio H/h = 2, and a parabolic inflow (average inflow velocity = 1). The cavity flow grid is 9 x 9 and the step flow grid is 21 x 9. The meshes and the velocity vectors are shown. y///#///////////////////////// Figure 2.3. Model 3 x 3 computational domain with numbered control volumes, for discussion of Eq. 2.22. The staggered velocity components which refer to control volume 5 are also indicated. I I I OP 7 OP P9 5 Un 5 P 4 0 Pr5 OP'6 s5 S -, - Pi P'2 P 3 Re = 100 Back-Step Flow 0 S-2 0 -4 0 O -6 S-6 ed) 2_ 100 200 300 # of Iterations 0 500 1000 # of Iterations Figure 2.4. Outer-loop convergence paths for the Re = 100 lid-driven cavity and backward-facing step flows, using central-differencing for the convection terms. Leg- end: p' equation: u momentum equation: -.-.-.- r momentum equation. 1500 Re = 100 Cavity Flow Re = 100 Cavity Flow Re = 100 Back-Step Flow S0 -3 0oo 3 \ 3 \ 0 .1 -6 -6 0 10 20 30 0 10 20 30 # of Iterations # of Iterations Figure 2.5. Inner-loop convergence paths for the Re = 100 lid-driven cavity and backward-facing step flows. The vertical axis is the logo of the ratio of the current residual to the initial residual. Legend: p' equation: --- u momentum equation: .-.-.- momentum equation. Inner Loop for Cavity Flow 0.03 I 0.01 # of Iterations Inner Loop for Back-Step Flow 50 # of Iterations Figure 2.6. Variation of p' with inner-loop iterations. The dashed line is the value of p' at the lower-left control volume, while the solid line is the difference between P'lowerleft and Pupperrnght 0.02 1: Outer Loop Convergence Path - 81 200 0 100 200 # of Iterations 0 50 # of Iterations Figure 2.7. Outer-loop and inner-loop convergence paths of the p' equation for the backward-facing step model problem, with and without enforcing the continuity con- straint. (1) conservation of mass not enforced: (2) continuity enforced globally; (3) cavity flow. Inner-Loop Convergence Path ca 3 ya -1 (U (^ Q- , Ii i H c J II . = 3 SrJ = ! n ce 0i oC *- C - . I^ cc-. i " - a- *~ -=rey JC k i-r i :-"t ; T- ^" .ihC ^ i-. -: ' T=11 T= 15 T=20 T=32 T=oo Figure 2.9. Time-dependent flowfield for impulsively started backward-facing step flow, Re = 800. The domain has length L = 8. Streamfunction contours are plotted at several instants during the evolution to the steady-state, which is the last figure. Time-Evolution of Reattachment/Separation Locations 0 10 20 30 40 50 Time Figure 2.10. Time-dependent location of bottom wall reattachment point and top wall separation point for Re = 800 impulsively started backward-facing step flow. The curves for both L = 8 and L = 16 computations are shown; they overlap identically. U Velocity Profile at X = 7 For Re = 800 Back-Step Flow U(Y) V Velocity Profile at X = 7 For Re = 800 Back-Step Flow -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 V(Y) Figure 2.11. Comparison of u and v-component of velocity profiles at x = 7.0 for the L = 16 and L = 8 backward-facing step simulations at Re = 800, with central- differencing. (o) indicates the grid-independent FEM solution obtained by Gartling. The v profile is scaled up by 103. CHAPTER 3 EFFICIENCY AND SCALABILITY ON SIMD COMPUTERS The previous chapter considered an issue which was important because of its im- plications for the convergence rate in open boundary problems. The present chapter shifts gears to focus on the cost and efficiency of pressure-correction methods on SIMD computers. As discussed in Chapter 1, the eventual goal is to understand the indirect cost [23], i.e. the parallel run time, of such methods on SIMD computers, and how this cost scales with the problem size and the number of processors. The run time is just the number of iterations multiplied by the cost per iteration. This chapter considers the cost per iteration. 3.1 Background The discussion of SIMD computers in Chapter 1 indicated similarities in the general layout of such machines and in the factors which affect program performance. More detail is given in this section to better support the discussion of results. 3.1.1 Speedup and Efficiency Speedup S is defined as S = T (3.1) where Tp is the measured run time using np processors. In the present work T1 is the run time of the parallel algorithm on one processor, including both serial and parallel computational work, but excluding the front-end-to-processor and interpro- cessor communication. On a MIMD machine it is sometimes possible to actually time 53 the program on one processor, but each SIMD processor is not usually a capable serial computer by itself, so Ti must be estimated. The timing tools on the CM-2 and CM-5 are very sophisticated, and can separately measure the time elapsed by the processors doing computation, doing various kinds of communication, and doing nothing (waiting for an instruction from the front-end, which might be finishing up some serial work before it can send another code block). Thus, it is possible to make a reasonable estimate for T1. Parallel efficiency is the ratio of the actual speedup to the ideal (np), which reflects the overhead costs of doing the computation in parallel: E- S.actu T1/Tp (3.2) Sideal np If Tcomp is the time in seconds spent by each of the np processors doing useful work (computation), Tinter-proc is the time spent by the processors doing interprocessor communication, and Tfe-to-proc is the time elapsed through front-end-to-processor communication, then each of the processors is busy a total of Tcomp + Tinter-proc seconds and the total run time on multiple processors is Tcomp + Tinter-proc Tfe-to-proc seconds. Assuming that the parallelism is high, i.e. a high percentage of the virtual processors are not idle, a single processor would need npTcomp time to do the same work. Thus, T1 = npTcomp, and from Eq. 3.2 E can be expressed as 1 1 E = (3.3) 1 + (Tinter-proc + Tfe-to-proc) /Tcomp 1 + (Tcomm) Tcomp Since time is work divided by speed, E depends on both machine-related factors and the implementational factors through Eq. 3.3. High parallel efficiency is not neces- sarily a product of fast processors or fast communications considered alone, instead it is the relative speeds that are important, and the relative amount of communication and computation in the program. Consider the machine-related factors first. 3.1.2 Comparison Between CM-2. CM-5, and MP-1 A 32-node CM-5 with vector units, a 16k processor CM-2, and a 1k processor MP-1 were used in the present study. The CM-5 has 4 GBytes total memory, while the CM-2 has 512 Mbytes, and the MP-1 has 64 MBytes. The peak speeds of these computers are 4, 3.5, and 0.034 Gflops, respectively, in double precision. Per proces- sor, the peak speeds are 32, 7, and 0.033 Mflops, with memory bandwidths of 128, 25, and 0.67 Mbytes/s [67, 83]. Clearly these are computers with very different capa- bilities, even taking into account the fact that peak speeds, which are based only on the processor speed under ideal conditions, are not an accurate basis for comparison. In the CM-2 and CM-5 the front-end computers are Sun-4 workstations, while in the MP-1 the front-end is a Decstation 5000. From Eq. 3.3, it is clear that the relative speeds of the front-end computer and the processors are important. Their ratio determines the importance of the front-end-to-processor type of communication. On the CM-2 and MP-1, there is just one of these intermediate processors, called either a. sequencer or an array control unit, respectively, while on the 32-node CM-5 the 32 SPARC microprocessors have the role of sequencers. Each SPARC node broadcasts to four vector units (VUs) which actually do the work. Thus a 32-node CM-5 has 128 independent processors. In the CM-2 the "pro- cessors" are more often called processing elements (PEs), because each one consists of a floating-point unit coupled with 32 bit-serial processors. Each bit-serial processor is the memory manager for a single bit of a 32-bit word. Thus, the 16k-processor CM-2 actually has only 512 independent processing elements. This strange CM-2 processor design came about basically as a workaround which was introduced to im- prove the memory bandwidth for floating-point calculations [66]. Compared to the CM-5 VUs, the CM-2 processors are about one-fourth as fast, with larger overhead costs associated with memory access and computation. The MP-1 has 1024 4-bit processors-compared to either the CM-5 or CM-2 processors, the MP-1 processors are very slow. The generic term "processing element" (PE), which is used occassion- ally in the discussion below, refers to either one of the VUs, one of the 512 CM-2 processors, or one of the MP-1 processors, whichever is appropriate. For the present study, the processors are either physically or logically imagined to be arranged as a 2-d mesh, which is a layout that is well-supported by the data networks of each of the computers. The data network of the 32-node CM-5 is a fat tree of height 3, which is similar to a binary tree except the bandwidth stays constant upwards from height 2 at 160 MBytes/s (details in [83]). One can expect approximately 480 MBytes/s for regular grid communication patterns (i.e. between nearest-neighbor SPARC nodes) and 128 MBytes/s for random (global) communica- tions. The randomly-directed messages have to go farther up the tree, so they are slower. The CM-2 network (a hypercube) is completely different from the fat-tree net- work and its performance for regular grid communication between nearest-neighbor processors is roughly 350 MBytes/s [67]. The grid network on the CM-2 is called NEWS (North-East-West-South). It is a subset of the hypercube connections se- lected at run time. The MP-1 has two networks: regular communications use X-Net (1.25 GBytes/s, peak) which connects each processor to its eight nearest neighbors, and random communications use a 3-stage crossbar (80 MBytes/s, peak). To summarize the relative speeds of these three SIMD computers it is sufficient for the present study to observe that the MP-1 has very fast nearest-neighbor com- munication compared to its computational speed, while the exact opposite is true for the CM-2. The ratio of nearest-neighbor communication speed to computation speed is smaller still for the CM-5 than the CM-2. Again, from Eq. 3.3, one expects that these differences will be an important factor influencing the parallel efficiency. 3.1.3 Hierarchical and Cut-and-Stack Data Mappings When there are more array elements (grid points) than processors, each processor handles multiple grid points. Which grid points are assigned to which processors is determined by the "data-mapping," also called the data layout. The processors repeat any instructions the appropriate number of times to handle all the array elements which have been assigned to it. A useful idealization for SIMD machines, however, is to pretend there are always as many processors as grid points. Then one speaks of the "virtual processor" ratio (VP) which is the number of array elements assigned to each physical processor. The way the data arrays are partitioned and mapped to the processors is a main concern for developing a parallel implementation. The layout of the data determines the amount of communication in a given program. When the virtual processor ratio is 1, there are an equal number of processors and array elements and the mapping is just one-to-one. When VP > 1 the mapping of data to processors is either "hierarchical," in CM-Fortran, or "cut-and-stack" in MP-Fortran. These mappings are also termed "block" and "cyclic" [85], respectively, in the emerging High-Performance Fortran standard. The relative merits of these different approaches have not been completely explored yet. In cut-and-stack mapping, nearest-neighbor array elements are mapped to nearest- neighbor physical processors. When the number of array elements exceeds the num- ber of processors, additional memory layers are created. VP is just the number of memory layers. In the general case, nearest-neighbor virtual processors (i.e. array elements) will not be mapped to the same physical processor. Thus, the cost of a nearest-neighbor communication of distance one will be proportional to VP, since the nearest-neighbors of each virtual processor will be on a different physical processor. In the hierarchical mapping, contiguous pieces of an array ("virtual subgrids") are mapped to each processor. The "subgrid size" for the hierarchical mapping is syn- onymous with VP. The distinction between hierarchical and cut-and-stack mapping is clarified by Figure 3.1. In hierarchical mapping, for VP > 1, each virtual processor has nearest-neighbors in the same virtual subgrid, that is, on the same physical processor. Thus, for hier- archical mapping on the CM-2, interprocessor communication breaks down into two types (with different speeds)-on-processor and off-processor. Off-processor commu- nication on the CM-2 has the NEWS speed given above, while on-processor communi- cation is somewhat faster, because it is essentially just a memory operation. A more detailed presentation and modelling of nearest-neighbor communication costs for the hierarchical mapping on the CM-2 is given in [3]. The key idea is that with hierar- chical mapping on the CM-2 the relative amount of on-processor and off-processor communication is the area to perimeter ratio of the virtual subgrid. For the CM-5, there are three types of interprocessor communication: (1) between virtual processors on the same processor (that is, the same VU), (2) between virtual processors on different VUs but on the same SPARC node, and (3) between virtual processors on different SPARC nodes. Between different SPARC nodes (number 2), the speed is 480 MBytes/s as mentioned above. On the same VU the speed is 16 GBytes/s. (The latter number is just the aggregate memory bandwidth of the 32- node CM-5.) Thus, although off-processor NEWS communication is slow compared to computation on the CM-2 and CM-5, good efficiencies can still be achieved as a consequence of the data mapping which allows the majority of communication to be of the on-processor type. 3.2 Implementional Considerations The cost per SIMPLE iteration depends on the choice of relaxation method (solver) for the systems of equations, the number of inner iterations (,,, v,, and vc), the computation of coefficients for each system of equations, the correction step, and the convergence checking and serial work done in program control. The pressure- correction equation, since it is not underrelaxed, typically needs to be given more iterations than the momentum equations, and consequently most of the effort is ex- pended during this step of the SIMPLE method. This is another reason why the convergence rate of the p'-equations discussed in Chapter 2 is important. Typically v. and v, are the same and are < 3, and v, < 5v,. In developing a parallel implementation of the SIMPLE algorithm, the first con- sideration is the method of solving the u, v, and p' systems of equations. For serial computations, successive line-underrelaxation using the tridiagonal matrix algorithm (TDMA, whose operation count is O(N)) is a good choice because the cost per it- eration is optimal and there is long-distance coupling between flow variables (along lines), which is effective in promoting convergence in the outer iterations. The TDMA is intrinsically serial. For parallel computations, a parallel tridiagonal solver must be used (parallel cyclic reduction in the present work). In this case the cost per it- eration depends not only on the computational workload (O(Nlog2N)) but also on the amount of communication generated by the implementation on a particular ma- chine. For these reasons, timing comparisons are made for several implementations of both point- and line-Jacobi solvers used during the inner iterations of the SIMPLE algorithm. Generally, point-Jacobi iteration is not sufficiently effective for complex flow prob- lems. However, as part of a multigrid strategy, good convergence rates can be ob- tained (see Chapters 4 and 5). Furthermore, because it only involves the fastest type of interprocessor communication, that which occurs between nearest-neighbor pro- cessors, point-Jacobi iteration provides an upper bound for parallel efficiency, against which other solvers can be compared. The second consideration is the treatment of boundary computations. In the present implementation, the coefficients and source terms for the boundary control volumes are computed using the interior control volume formula and mask arrays. Oran et al. [57] have called this trick the uniform boundary condition approach. All coefficients can be computed simultaneously. The problem with computing the boundary coefficients separately is that some of the processors are idle, which de- creases E. For the CM-5, which is "synchronized MIMD" instead of strictly SIMD, there exists limited capability to handle both boundary and interior coefficients si- multaneously without formulating a single all-inclusive expression. However, this capability cannot be utilized if either the boundary or interior formulas involve in- terprocessor communication, which is the case here. As an example of the uniform approach, consider the source terms for the north boundary u control volumes, which are computed by the formula b = aNUN + (pw Pe)Ay (3.4) Recall that aN represents the discretized convective and diffusive flux terms, and UN is the boundary value, and in the pressure gradient term, Ay is the vertical dimension of the u control volume and pw/P are the west/east u-control-volume face pressures on the staggered grid. Similar modifications show up in the south, east, and west boundary u control volume source terms. To compute the boundary and interior source terms simultaneously, the following implementation is used: b = aboundaryUboundary + (Pw pe)Ay (3.5) where Uboundary = UNIN + USIS + UEIE + UWIW (3.6) and boundary = aNIN + asls + aEIE + awIw (3.7) IN, Is, IE, and Iw are the mask arrays, which have the value 1 for the respective boundary control volumes and 0 everywhere else. They are initialized once, at the beginning of the program. Then, every iteration, there are four extra nearest-neighbor communications. A comparison of the uniform approach with an implementation that treats each boundary separately is discussed in the results. 3.3 Numerical Experiments The SIMPLE algorithm for two-dimensional laminar flow has been timed on a range of problem sizes from 8 x 8 to 1024 x 1024 which, on the CM-5, covers up to VP = 8192. The convection terms are central-differenced. A fixed number (100) of outer iterations are timed using as a model flow problem the lid-driven cavity flow at Re = 1000. The timings were made with the "Prism" timing utility on the CM-2 and CM-5, and the "dpuTimer" routines on the MP-1 [52, 86]. These utilities can be inaccurate if the front-end machine is heavily loaded, which was the case with the CM-2. Thus, on the CM-2 all cases were timed three times and the fastest times were used, as recommended by Thinking Machines [82]. Prism times every code block and accumulates totals in several categories, including computation time for the nodes (Tcomp), "NEWS" communication (Tnews), and irregular-pattern "SEND" communication. Also it is possible to infer Tfe-to-proc from the difference between the processor busy time and the elapsed time. In the results Tcomm is the sum of the "NEWS" and "SEND" interprocessor times. The front-end-to-processor communication is separate. Additionally, the component tasks of the algorithm have been timed, namely the coefficient computations (Tcoe/,), the solver (Tsoi,,), and the velocity-correction and convergence-checking parts. 3.3.1 Efficiency of Point and Line Solvers for the Inner Iterations Figure 3.2, based on timings made on the CM-5, illustrates the difference in parallel efficiency for SIMPLE using point-Jacobi and line-Jacobi iterative solvers. E is computed from Eq. 3.3 by timing Tcomm and Teomp introduced above. Problem size is given in terms of the virtual processor ratio VP previously defined. There are two implementations each with different data layouts, for point-Jacobi iteration. One ignores the distinction between virtual processors which are on the same physical processor and those which are on different physical processors. Each array element is treated as if it is a processor. Thus, interprocessor communication is generated whenever data is to be moved, even if the two virtual processors do- ing the communication happen to be on the same physical processor. To be more precise, a call to the run-time communication library is generated for every array el- ement. Then, those array elements (virtual processors) which actually reside on the same physical processor are identified and the communication is done as a memory operation-but the unnecessary overhead of calling the library is incurred. Obviously there is an inefficiency associated with pretending that there are as many processors as array elements, but the tradeoff is that this is the most straightforward, and indeed the intended, way to do the programming. In Figure 3.2, this approach is labelled "NEWS," with the symbol "o." The other implementation is labelled "on-VU," with the symbol "+," to indicate that interprocessor communication between virtual pro- cessors on the same physical processor is being eliminated-the programming is in a sense being done "on-VU." To indicate to the compiler the different layouts of the data which are needed, the programmer inserts compiler directives. For the "NEWS" version, the arrays are laid out as shown in this example for a 1k x 1k grid and an 8 x 16 processor layout on the CM-5: REAL*8 A(1024,1024) $CMF LAYOUT A(:BLOCK=128 :PROCS=8, :BLOCK=64 :PROCS=16) Thus, the subgrid shape is 128 x 64, with a subgrid size (VP) of 8192 (this hap- pens to be the biggest problem size for my program on a 32-node CM-5 with 4GBytes of memory). When shifting all the data to their east nearest-neighbor, for example, by far the large majority of transfers are on-VU and could be done without real inter- processor communication. But there are only 2 dimensions in A, so that data-parallel program statements cannot specifically access certain array elements, i.e. the ones on the perimeter of the subgrid. Thus it is not possible with the "NEWS" layout to treat interior virtual processors differently from those on the perimeter, and conse- quently data shifts between the interior virtual processors generate interprocessor communication even though it is unnecessary. In the "on-VU" version, a different data layout is used which makes explicit to the compiler the boundary between physical processors. The arrays are laid out without virtual processors: $CMF LAYOUT A(:SERIAL,:SERIAL,:BLOCK=1 :PROCS=8,:BLOCK=1 :PROCS=16) The declaration must be changed accordingly, to A(128,64,8,16). Normally it is inconvenient to work with the arrays in this manner. Thus the approach taken here is to use an "array alias" of A [84]. In other words, this is an EQUIVALENCE func- tion for the data-parallel arrays (similar to the Fortran77 EQUIVALENCE concept), which equates A(1024,1024) with A(128,64,8,16), with the different LAYOUTs given above. It is the alias instead of the original A which is used in the on-VU point- Jacobi implementation. In the solver, the "on-VU" layout is used; everywhere else, the more convenient "NEWS" layout is used. The actual mechanism by which the equivalencing of distributed arrays can be accomplished is not too difficult to under- stand. The front-end computer stores "array descriptors," which contain the array layout, the starting address in processor memory, and other information. The actual layout in each processors' memory is linear and doesn't change, but multiple array descriptors can be generated for the same data. This descriptor multiplicity is what array aliasing accomplishes. With the "on-VU" programming style, the compiler does not generate communication when the shift of data is along a SERIAL axis. Thus, interprocessor communication is generated only when the virtual processors involved are on different physical processors, i.e. only when it is truly necessary. The difference in the amount of communication is substantial for large subgrid sizes. For both the "NEWS" and the "on-VU" curves in Figure 3.2, E is initially very low, but as VP increases, E rises until it reaches a peak value of about 0.8 for the "NEWS" version and 0.85 for the "on-VU" version. The trend is due to the amor- tization of the front-end-to-processor and off-VU (between VUs which are physically under control of different SPARC nodes) communication. The former contributes a constant overhead cost per Jacobi iteration to Tcomm, while the latter has a VP1/2 dependency [3]. However, it does not appear from Figure 3.2 that these two terms' effects can be distinguished from one another. For VP > 2k, the CM-5 is computing roughly 3/4 of the time for the implementa- tion which uses the "NEWS" version of point-Jacobi, with the remainder split evenly between front-end-to-processor communication and on-VU interprocessor communi- cation. It appears that the "on-VU" version has more front-end-to-processor com- munication per iteration, so there is, in effect, a price of more front-end-to-processor communication to pay in exchange for less interprocessor communication. Conse- quently it takes VP > 4k to reach peak efficiency instead of 2k with the "NEWS" version. For VP > 4k, however, E is about 5% 10% higher than for the "NEWS" version because the on-VU communication has been replaced by straight memory operations. The observed difference would be even greater if a larger part of the total parallel run time was spent in the solver. For the large VP cases in Figure 3.2, approximately equal time was spent computing coefficients and solving the systems of equations. "Typical" numbers of inner iterations were used, 3 each for the u and v momentum equations, and 9 for the p' equation. From Figure 3.2, then, it appears that the ad- vantage of the "on-VU" version over the "NEWS" version of point-Jacobi relaxation within the SIMPLE algorithm is around 0.1 in E, for large problem sizes. Red/black analogues to the "NEWS" and "on-VU" versions of point-Jacobi iter- ation have also been tested. Red/black point iteration done in the "on-VU" manner does not generate any additional front-end-to-processor communication, and there- fore takes almost an identical amount of time as point-Jacobi. Thus red/black point iterations are recommended when the "on-VU" layout is used due to their improved convergence rate. However, with the "NEWS" layout, red/black point iteration gen- erates two code blocks instead of one, and reduces by 2 the amount of computation per code block. This results in a substantial (- 35% for the VP = 8k case) in- crease in run time. Thus, if using "NEWS" layouts, red/black point iteration is not cost-effective. There are also two implementations of line-Jacobi iteration. In both, one inner iteration consists of forming a tridiagonal system of equations for the unknowns in each vertical line by moving the east/west terms to the right-hand side, solving the multiple systems of equations simultaneously, and repeating the procedure for the horizontal lines. In the first version, parallel cyclic reduction is used to solve the multiple tridiag- onal systems of equations (see [44] for a clear presentation). This involves combining equations to decouple the system into even and odd equations. The result is two tridiagonal systems of equations each half the size of the original. The reduction step is repeated log2 N times, where N is the number of unknowns in each line. Thus, the computational operation count is O(Nlog2N). Interprocessor communication occurs for every unknown for every step, thus the communication operation count is also O(Nlog2N). However, the distance for communication increases every step of the re- duction by a factor of 2. For the first step, nearest-neighbor communication occurs, while for the second step, the distance is 2, then 4, etc. Thus, the net communi- cation speed is slower than the nearest-neighbor type of communication. Figure 3.2 confirms this argument-E peaks at about 0.5 compared to 0.8 for point-Jacobi it- eration. In other words, for VP > 4k, interprocessor communication takes as much time as computation with the line-Jacobi solver using cyclic reduction. In the second version, the multiple systems of tridiagonal equations are solved using the standard TDMA algorithm along the lines. To implement this version, one must remap the arrays from (:NEWS,:NEWS) to (:NEWS,:SERIAL), for the vertical lines, and to (:SERIAL,:NEWS) for the horizontal lines. This change from rectangular subgrids to 1-d slices is the most time-consuming step, involving a global communication of data ("SEND" instead of "NEWS"). Applied along the serial di- mension, the TDMA does not generate any interprocessor communication. Some front-end-to-processor communication is generated by the incrementing of the DO- loop index, but unrolling the DO-loop helps to amortize this overhead cost to some extent. Thus, in Figure 3.2 E is approximately constant at 0.14, except for very small VP. The global communication is much slower than computation and consequently there is not enough computation to amortize the communication. Furthermore, the constant E implies from Eq. 3.3 that Tcomm and Teomp both scale in the same way with problem size. It is evident that Tcomp ~ VP because the TDMA is O(N). Thus constant E implies Tcomm VP. This means doubling VP doubles Tcomm, indicating the communication speed has reached its peak, which further indicates that the full bandwidth of the fat-tree is being utilized. The disappointing performance of the standard line-iterative approach using the TDMA points out the important fact that, for the CM-5, global communication within inner iterations is intolerable. There is not enough computation to amortize slow communication in the solver for any problem size. With parallel cyclic reduction, where the regularity of the data movement allows faster communication, the efficiency is much higher, although still significantly lower than for point-iterations. Additional improvement can be sought by using the "on-VU" data layout to implement the line-iterative solver within each processor's subgrid. This implementation essentially trades interprocessor communication for the front-end-to-PE type of communication, and in practice a front-end bottleneck develops. For the remainder of the discussion, all line-Jacobi results refer to the parallel cyclic reduction implementation. On the MP-1, the front-end-to-processor communication is not a major concern, as can be inferred from Figure 3.3. The efficiency of the SIMPLE algorithm using the point-Jacobi solver is plotted for each machine for the range of problem sizes corresponding to the cases solved on the MP-1. The CM-2 and CM-5 can solve much larger problems, so for comparison purposes only part of their data is shown. Also, because the computers have different numbers of processors, the number of grid points is used instead of VP to define the problem size. As in Figure 3.2, each curve exhibits an initial rise corresponding to the amortiza- tion of the front-end-to-processor communication and, for the CM-2 and CM-5, the off-processor "NEWS" communication. On the MP-1, peak E is reached for small problems (VP > 32). Due to the MP-l's relatively slow processors, the computa- tion time quickly amortizes the front-end-to-processor communication time as VP increases. Furthermore, because the relative speed of X-Net communication is fast, the peak E is high, 0.85. On the CM-2, the peak E is 0.4, and this efficiency is reached for approximately VP > 128. On the CM-5, the peak E is 0.8, but this efficiency is not reached until VP > 2k. If computation is fast, then the rate of in- crease of E with VP depends on the relative cost of on-processor, off-processor, and front-end-to-processor communication. If the on-processor communication is fast, larger VP is required to reach peak E. Thus, on the CM-5, the relatively fast on-VU communication is simultaneously responsible for the good (0.8) peak E, and the fact that very large problem sizes, (VP > 2k, 64 times larger than on the MP-1), are needed to reach this peak E. The aspect ratio of the virtual subgrid constitutes a secondary effect of the data layout on the efficiency for hierarchical mapping. The major influence on E depends on VP, i.e. the subgrid size, but the subgrid shape matters, too. This dependence comes into play due to the different speeds of the on-processor and off-processor types of communication. Higher aspect ratio subgrids have higher area to perimeter ratios, and thus relatively more of off-processor communication than square subgrids. Figure 3.4 gives some idea of the relative importance of the subgrid aspect ratio effect. Along each curve the number of grid points is fixed, but the grid dimensions vary, which, for a given processor layout, causes the subgrid shape (aspect ratio), to vary. For example, on the CM-5 with an 8 x 16 processor layout, the following grids were used corresponding to the VP = 1024 CM-5 curve: 256 x 512, 512 x 256, 680 x 192, and 1024 x 128. These cases give subgrid aspect ratios of 1, 4, 7, and 16. Tnews is the time spent in "NEWS" type of interprocessor communication and Tcom, is the time spent doing computation during 100 SIMPLE iterations. The solver for these results is point-Jacobi relaxation. For the VP = 1024 CM-5 case, increasing the aspect ratio from 1 to 16 causes Tnews/Tcomp to increase from 0.3 to 0.5. This increase in Tnews/Tcomp increases the run time for 100 iterations from 15s to 20s, and decreases the efficiency from 0.61 to 0.54. For the VP = 8192 CM-5 case, increasing the aspect ratio from 1 to 16 causes Tnews/Tcomp to increase from 0.19 to 0.27. This increase in Tnews/Tcomp increases the run time for 100 iterations from 118s to 126s, and decreases the efficiency from 0.74 to 0.72. Thus, the aspect ratio effect diminishes as VP increases due to the increasing area of the subgrid. In other words the variation in the perimeter length matters less, percentage-wise, as the area increases. The CM-2 results are similar. However, on the CM-2 the on-PE type of communication is slower than on the CM-5, relative to the computational speed. Thus, Te,,,/Tcomp ratios are higher on the CM-2. 3.3.2 Effect of Uniform Boundary Condition Implementation In addition to the choice of solver, the treatment of boundary coefficient computa- tions was discussed earlier as an important consideration affecting parallel efficiency. Figure 3.5 compares the implementation described in the introductory section of this chapter, to an implementation which treats the boundary control volumes separate from the interior control volumes. The latter approach involves some 1-d operations which leave some processors idle. The results indicated in Figure 3.5 were obtained on the CM-2, using point- Jacobi relaxation as the solver. With the uniform approach, the ratio of the time spent computing coefficients, Tcoeff, to the time spent solving the equations, Tsolve, remains constant at 0.6 for VP > 256. Both Tcoeff and To ,,ve VP in this case, so doubling VP doubles both Tcoeff and To,,ve, leaving their ratio unchanged. The value 0.6 reflects the relative cost of coefficient computations compared to point-Jacobi iteration. There are three equations for which coefficients are computed and 15 total inner iterations, 3 each for the u and v equations, and 9 for the p' equation. Thus if more inner iterations are taken, the ratio of Tcoeff to Tsoive will decrease, and vice- versa. With the 1-d implementation, Tcoeff/Tsove increases until VP > 1024. Both TcoeJf and Tso,,ve scale with VP asymptotically, but Figure 3.5 shows that Tcoeff has an apparently very significant square-root component due to the boundary operations. If N is the number of grid points and n, is the number of processors, then VP = N/np. For boundary operations, N1/2 control volumes are computed in parallel with only nt/2 processors-hence the VP1/2 contribution to Tcoeff. From Figure 3.5, it appears that very large problems are required to reach the point where the interior coefficient computations amortize the boundary coefficient computations. Even for large VP when Tcoeff/Tsoive is approaching a constant, this constant is larger, approximately 0.8 compared to 0.6 for the uniform approach, due to the additional front-end-to- processor communication which is intrinsic to the 1-d formulation. 3.3.3 Overall Performance Table 3.1 summarizes the relative performance of SIMPLE on the CM-2, CM-5, and MP-1 computers, using point and line-iterative solvers and the uniform boundary condition treatment. In the first three cases the "NEWS" implementation of point- Jacobi relaxation is the solver, while the last two cases are for the line-Jacobi solver using cyclic reduction. Machine Solver Problem VP Tp Time/Iter./Pt. Speed ) Peak Size ___(MFlops) Speed 512 PE Point- 512 x 1024 188 s 2.6 x 10-6 s 147 4 CM-2 Jacobi 1024 128 VU Point- 736 x 8192 137 s 1.3 x 10-6 s 417 10 CM-5 Jacobi 1472 1024 PE Point- 512 x 256 316 s 1.2 x 10-5 s 44* 59 MP-1 Jacobi 512 512 PE Line- 512 x 1024 409 s 7.8 x 10-6 s 133 3 CM-2 Jacobi 1024 128 VU Line- 736 x 8192 453 s 4.2 x 10-6 s 247 6 CM-5 Jacobi 1472 Table 3.1. Performance results for the SIMPLE algorithm for 100 iterations of the model problem. The solvers are the point-Jacobi ("NEWS") and line-Jacobi (cyclic reduction) implementations. 3, 3, and 9 inner iterations are used for the u, v, and p' equations, respectively. *The speeds are for double-precision calculations, except on the MP-1. In Table 3.1, the speeds reported are obtained by comparing the timings with the identical code timed on a Cray C90, using the Cray hardware performance mon- itor to determine Mflops. In terms of Mflops, the CM-2 version of the SIMPLE algorithm's performance appears to be consistent with other CFD algorithms on the CM-2. Jesperson and Levit [44] report 117 Mflops for a scalar implicit version of an approximate factorization Navier-Stokes algorithm using parallel cyclic reduction to solve the tridiagonal systems of equations. This result was obtained for a 512 x 512 simulation of 2-d flow over a cylinder using a 16k CM-2 as in the present study (a different execution model was used (see [3, 47] for details). The measured time per time-step per grid point was 1.6 x 10-5 seconds. By comparison, the performance of the SIMPLE algorithm for the 512 x 1024 problem size using the line-Jacobi solver is 133 Mflops and 7.8 x 10-6 seconds per iteration per grid pt. Egolf [20] reports that the TEACH Navier-Stokes combustor code based on a sequential pressure-based method with a solver that is comparable to point-Jacobi relaxation, obtains a performance which is 3.67 times better than a vectorized Cray X-MP version of the code, for a model problem with 3.2 x 104 nodes. The present program runs 1.6 times faster than a single Cray C90 processor for a 128 x 256 problem (32k grid points). One Cray C-90 processor is about 2-4 times faster than a Cray X-MP. Thus, the present code runs comparably fast. 3.3.4 Isoefficiencv Plot Figures 3.2-3.4 addressed the effects of the inner-iterative solver, the boundary treatment, the data layout, and the variation of parallel efficiency with problem size for a fixed number of processors. Varying the number of processors is also of interest and, as discussed in Chapter 1, an even more practical numerical experiment is to vary np in proportion with the problem size, i.e. the scaled-size model. Figure 3.6, which is based on the point-Jacobi MP-1 timings, incorporates the above information into one plot, which has been called an isoefficiency plot by Kumar and Singh [46]. The lines are paths along which the parallel efficiency E remains constant as the problem size and the number of processors np vary. Using the point- Jacobi solver and the uniform boundary coefficient implementation, each SIMPLE iteration has no substantial contribution from operations which are less than fully parallel or from operations whose time depends on the number of processors. The efficiency is only a function of the virtual processor ratio, thus the lines are straight. Much of the parameter space is covered by efficiencies between 0.6 and 0.8. The reason that the present implementation is linearly scalable is that the oper- ations are all scalable-each SIMPLE iteration has predominantly nearest-neighbor communication and computation and full parallelism. Thus, Tp depends on VP. Local communication speed does not depend on np. Ti depends on the problem size N. Thus, as N and n, are increased in proportion, starting from some initial ratio, the efficiency from Eq. 3.3 stays constant. If the initial problem size is large and the corresponding parallel run time is acceptable, then one can quickly get to very large problem sizes while still maintaining Tp constant by increasing n, a relatively small amount (along the E = 0.85 curve). If the desired run time is smaller, then initially (i.e. starting from small np) the efficiency will be lower. Then the scaled-size experiment requires relatively more processors to get to a large problem size along the constant efficiency (constant Tp for point-Jacobi ierations) curve. Thus, the most desirable situation occurs when the efficiency is high for an initially small problem size. For this case the fixed-time and scaled-size methods are equivalent, because the problem size T1 depends on N per iteration. However this is not the case when the SIMPLE inner iterations are done with the line-Jacobi solver using parallel cyclic reduction. Cyclic reduction requires (13 log2 N+1)N operations to solve a tridiagonal system of N equations [44]. Thus, T ~- (13log2 N + 1)N and on np = N processors, Tp ~ 13 log2 N+ 1 because every processor is active during every step of the reduction and there are 13 log2 N 1 steps. Since VP = 1, every processor's time is proportional to the number of steps, assuming each step costs about the same. In the scaled-size approach, one doubles np and N together, which therefore gives Ti ~ (26 log, 2N+2)N and Tp 13 log, 2N+1. The efficiency is 1, but Tp is increased and T1 is more than doubled. In the fixed-time approach, then, one concludes that N must be increased by a factor which is less than two, and n, must be doubled, in order to maintain constant Tp. If a plot like Figure 3.6 is constructed, it should be done with Ti instead of N as the measure of problem size. In that case, the lines of constant efficiency would be described as Ti ~ n0, with a > 1. The ideal case is a = 1. In addition to the operation count, there is another factor which reduces the scalability of cyclic reduction, namely the time per step is not actually the same as was assumed above-later steps require communication over longer distances which is slower. In practice, however, no more than a few steps are necessary because the coupling between widely-separated equations becomes very weak. As the system is reduced the diagonal becomes much larger than the off-diagonal terms which can then be neglected and the reduction process abbreviated. In short, the basic prerequisite for scaled-size constant efficiency is that the amount of work per SIMPLE iteration varies with VP and that the overheads and inefficiencies, specifically the time spent in communication and the fraction of idle processors, do not grow relative to the useful computational work as np and N are increased proportionally. The SIMPLE implementation developed here using the point-iterative solvers, Jacobi and red/black, have this linear computational scalabil- ity property. On the other hand, the convergence rate of point-iterative methods increases at a rate greater than the problem size, so although Tp can be maintained constant while the problem size and np are scaled up, the convergence rate deteriorates. Hence the total run time (cost per iteration multiplied by the number of iterations) increases. This lack of numerical scalability of standard iterative methods like point-Jacobi relaxation is the motivation for the development of multigrid strategies. 3.4 Concluding Remarks The SIMPLE algorithm, especially using point-iterative methods, is efficient on SIMD machines and can maintain a relatively high efficiency as the problem size and the number of processors is scaled up. However, boundary coefficient computations need to be folded in with interior coefficient computations to achieve good efficiencies at smaller problem sizes. For the CM-5, the inefficiency caused by idle processors in a 1-d boundary treatment was significant over the entire range of problem sizes tested. The line-Jacobi solver based on parallel cyclic reduction leads to a lower peak E (0.5 on the CM-5) than the point-Jacobi solver (0.8), because there is more communication and on average this communication is less localized. On the other hand, the asymptotic convergence rates of the two methods are also different and need to be considered on a problem-by-problem basis. The speeds which are obtained with the line-iterative method are consistent and comparable with other CFD algorithms on SIMD computers. The key factor in obtaining high parallel efficiency for the SIMPLE algorithm on the computers used, is fast nearest-neighbor communication relative to the speed of computation. On the CM-2 and CM-5, hierarchical mapping allows on-processor communication to dominate the slower off-processor form(s) of communication for large VP. The efficiency is low for small problems because of the relatively large contribution to the run time from the front-end-to-processor type of communication, but this type of communication is constant and becomes less important as the problem size increases. Once the peak E is reached, the efficiency is determined by the balance of compu- tation and on-processor communication speeds-for the CM-5, using a point-Jacobi solver, E approaches approximately 0.8, while on the CM-2 the peak efficiency is 0.4, which reflects the fact that the CM-5 vector units have a better balance, at least for the operations in this algorithm, than the CM-2 processors. The rate at which E approaches the peak value depends on the relative contribu- tions of on- and off-processor communication and front-end-to-processor communica- tion to the total run time. On the CM-5, VP > 2k is required to reach peak E. This problem size is about one-fourth the maximum size which can be accommodated, and yet still larger than many computations on traditional vector supercomputers. Clearly a gap is developing between the size of problems which can be solved effi- ciently in parallel and the size of problems which are small enough to be solved on serial computers. For parallel computations of all but the largest problems, then, the data layout issue is very important- in going from a square subgrid to one with aspect ratio of 16, for a VP = 1k case on the CM-5, the run time increased by 25%. On the MP-1, hierarchical mapping is not needed, because the processors are slow compared to the X-Net communication speed. The peak E is 0.85 with the point-Jacobi solver, and this performance is obtained for VP > 32, which is about one-eighth the size of the largest case possible for this machine. Thus, with regards to achieving efficient performance in the teraflops range, the comparison given here suggests a preference for numerous slow processors instead of fewer fast ones, but such a computer may be difficult and expensive to build. 4 x 1 Layout of Processors PE 0 PE 1 PE 2 PE 3 Array A(8) 1 21 31 61 Cut-and-Stack Mapping (MP-Fortran) Memory Layers i/ 5/ 6/ 7/ 8/ E 1/2 / 3PE 4/ PE0 PE1 PE2 PE3 Hierarchical Mapping (CM-Fortran) 2 x1 virtual subgrids 1 2 3 4 5 6 7 8 PE 0 PE 1 PE 2 PE 3 Figure 3.1. Mapping an 8 element array A onto 4 processors. For the cut-and- stack mapping, nearest-neighbors array elements are mapped to nearest-neighbor physical processors. For the hierarchical mapping, nearest-neighbor array elements are mapped to nearest-neighbor virtual processors, which may be on the same physical processor. 7 1 81 Efficiency vs. VP wU 1 0.8 0.6 0.4 0.2, 1 0' 0 10000 5000 VP Figure 3.2. Parallel efficiency, E, as a function of problem size and solver, for the CM-5 cases. The number of grid points is the virtual processor ratio, VP, multiplied by the number of processors, 128. E is computed from Eq. 3.3. It reflects the relative amount of communication, compared to computation, in the algorithm. + + 0o 0 0 m + Point-Jacobi (on-VU) o Point-Jacobi (NEWS) Line-Jacobi (Cyclic Red.) x Line-Jacobi (TDMA) txx x x x XX ) E vs. Problem Size LL 1 0.8 0.6 0.4 0.2 0 C 1 2 # of Grid Points x 10 5 Figure 3.3. Comparison between the CM-2, CM-5 and MP-1. The variation of parallel efficiency with problem size is shown for the model problem, using point- Jacobi relaxation as the solver. E is calculated from Eq. 3.3, and T1 = nrpTcom for the CM-2 and CM-5, where Tcomp is measured. For the MP-1 cases, T1 is the front-end time, scaled down to the estimated speed of the MP-1 processors (0.05 Mflops). 0 0 0 0 D + ) + X X o MP-1 S+ CM-5 CM-2 ) 80 Aspect Ratio Effect 2 E 1.5 O + VP=256, CM-2 0 o VP=1024, CM-2 1 VP=1024, CM-5 x VP=8192, CM-5 0.5 - 0 |-0-------------- 0 5 10 15 20 Subgrid AR Figure 3.4. Effect of subgrid aspect ratio on interprocessor communication time, T,ew, for the hierarchical data-mapping (CM-2 and CM-5). Tnew, is normalized by Tcom, in order to show how the aspect ratio effect varies with problem size, without the complication of the fact that Tcomp varies also. Effect of Implementation 0.8 0.75 0 c)0.7 So 1-d operations S0.65 + 2-d operations 0 0 0.6 ---- 0.55 0 500 1000 1500 VP Figure 3.5. Normalized coefficient computation time as a function of problem size, for two implementations (on the CM-2). In the 1-d case the boundary coefficients are handled by 1-d array operations. In the 2-d case the uniform implementation computes both boundary and interior coefficients simultaneously. Tcoeff is the time spent computing coefficients in a SIMPLE iteration; Toie, is the time spent in point- Jacobi iterations. There are 15 point-Jacobi iterations (v, = v, = 3 and v, = 9). Isoefficiency Curves in -o x CD 0 IL 0C Vf 2.5 2 1.5 1 0.5 2000 4000 6000 8000 # Processors (MP-1) Figure 3.6. Isoefficiency curves based on the MP-1 cases and SIMPLE method with the point-Jacobi solver. Efficiency E is computed from Eq. 3.3. Along lines of constant E the cost per SIMPLE iteration is constant with the point-Jacobi solver and the uniform boundary condition implementation. CHAPTER 4 A NONLINEAR PRESSURE-CORRECTION MULTIGRID METHOD The single-grid timing results focused on the cost per iteration in order to elucidate the computational issues which influence the parallel run time and the scalability. But the parallel run time is the cost per iteration multiplied by the number of iterations. For scaling to large problem sizes and numbers of processors, the numerical method must scale well with respect to convergence rate, also. The convergence rate of the single-grid pressure-correction method deteriorates with increasing problem size. This trait is inherited from the smoothing property of the stationary linear iterative method, point or line-Jacobi relaxation, used to solve the systems of u, v, and p' equations during the course of SIMPLE iterations. Point- Jacobi relaxation requires O(N2) iterations, where N is the number of grid points, to decrease the solution error by a specified amount [1]. In other words, the number of iterations increases faster than the problem size. At best the cost per iteration stays constant as the number of processors np increases proportional to the problem size. Thus, the total run time increases in the scaled-size experiment using single-grid pressure-correction methods, due to the increased number of iterations required. This lack of numerical scalability is a serious disadvantage for parallel implementations, since the target problem size for parallel computation is very large. Multigrid methods can maintain good convergence rates as the problem size in- creases. For Poisson equations, problem-size independent convergence rates can be obtained [36, 55]. The recent book by Briggs [10] introduces the major concepts in 83 the context of Poisson equations. See also [11, 37, 90] for surveys and analyses of multigrid convergence properties for more general linear equations. For a description of practical techniques and special considerations for fluid dynamics, see the impor- tant early papers by Brandt [5, 6]. However, there are many unresolved issues for application to the incompressible Navier-Stokes equations, especially with regards to their implementation and performance on parallel computers. The purpose of this chapter is to describe the relevant convergence rate and stability issues for multigrid methods in the context of application to the incompressible Navier-Stokes equations, with numerical experiments used to illustrate the points made, in particular, regard- ing the role of the restriction and prolongation procedures. 4.1 Background The basic concept is the use of coarse grids to accelerate the asymptotic con- vergence rate of an inner iterative scheme. The inner iterative method is called the "smoother" for reasons to be made clear shortly. In the context of the present applica- tion to the incompressible Navier-Stokes equations, the single-grid pressure-correction method is the inner iterative scheme. Because the pressure-correction algorithm also uses inner iterations-to solve the systems of u, v, and p' equations-the multigrid method developed here actually has three nested levels of iterations. A multigrid V cycle begins with a certain number of smoothing iterations on the fine grid, where the solution is desired. Figure 4.1 shows a schematic of a V(3,2) cycle. In this case three pressure-correction iterations are done first. Then residuals and variables are restricted (averaged) to obtain coarse-grid values for these quantities. The solution to the coarse-grid discretized equation provides a correction to the fine- grid solution. Once the solution on the coarse grid is obtained, the correction is interpolated (prolongated) to the fine grid and added back into the solution there. Some post-smoothing iterations, two in this case, are needed to eliminate errors introduced by the interpolation. Since it is usually too costly to attempt a direct solution on the coarse grid, this smoothing-correction cycle is applied recursively, leading to the V cycle shown. The next section describes how such a procedure can accelerate the convergence rate of an iterative method, in the context of linear equations. The multigrid scheme for nonlinear scalar equations and the Navier-Stokes system of equations is then described. Brandt [5] was the first to formalize the manner in which coarse grids could be used as a convergence-acceleration technique for a given smoother. The idea of using coarse grids to generate initial guesses for fine-grid solutions was around much earlier. The cost of the multigrid algorithm, per cycle, is dominated by the smoothing cost, as will be shown in Chapter 5. Thus, with regard to the parallel run time per multigrid iteration, the smoother is the primary concern. Also, with regard to the convergence rate, the smoother is important. The single-grid convergence rate characteristics of pressure-correction methods, the dependence on Reynolds number, flow problem, and the convection scheme, carry over to the multigrid context. However, in the multigrid method the smoother's role is, as the name implies, to smooth the fine-grid residual, which is a different objective than to solve the equations quickly. A smooth fine-grid residual equation can be approximated accurately on a coarser grid. The next section describes an alternate pressure-based smoother, and compares its cost against the pressure-correction method on the CM-5. Stability of multigrid iterations is also an important unresolved issue. There are two ways in which multigrid iterations can be caused to diverge. First, the single-grid smoothing iterations can diverge, for example if central-differencing is used there are possibly stability problems if the Reynolds number is high. Second, poor coarse-grid corrections can cause divergence if the smoothing is insufficient. In a sense this latter issue, the scheme and intergrid transfer operators which prescribe the coordination between coarse and fine grids in the multigrid procedure, is the key issue. In the next section two "stabilization strategies" are described. Then, the impact of different restriction and prolongation procedures on the convergence rate is studied in the context of two model problems, lid-driven cavity flow and flow past a symmetric backward-facing step. These two particular flow problems have different physical characteristics, and therefore the numerical experiments should give insight into the problem-dependence of the results. 4.1.1 Terminology and Scheme for Linear Equations The discrete problem to be solved can be written Ahuh = Sh, corresponding to some differential equation L[u] = S. The set of values uh is defined by {u}, = u(ih,jh), (i,j) E ([0 : N], [0 : N]) -- (4.1) Similarly, u2h is defined on the coarser grid f2h with grid spacing 2h. The variable u can be a scalar or a vector, and the operator A can be linear or nonlinear. For linear equations, the "correction scheme" (CS) is frequently used. A two- level multigrid cycle using CS accelerates the convergence of an iterative method (with iteration matrix P) by the following procedure: Do v fine-grid iterations vh PVvh Compute residual on Qh rh = Ahvh Sh Restrict rh to Q2h r2h = 2hrh Solve exactly for e2h e2h = (A2h)-1r2h Correct vh on Qh (h)new (vh)old + hhe2h Ih' and I,' symbolize the restriction and prolongation procedures. The quantity vh is the current approximation to the discrete solution uh. The algebraic error is the difference between them, eh = uh vh. The discretization error is the difference between the exact solutions of the continuous and discrete problems, ediscr = u uh. The truncation error is obtained by substituting the exact solution into the discrete equation, 7h = Ahu Sh = Ahu Ahuh. (4.2) The notation above follows Briggs [10]. The two-level multigrid cycle begins on the fine grid with v iterations of the smoother. Standard iterative methods all have the "smoothing property," which is that the various eigenvector-decomposed components of the solution error are damped at a rate proportional to their corresponding eigenvalues, i.e. the high frequency errors are damped faster than the low frequency (smooth) errors. Thus, the conver- gence rate of the smoothing iterations is initially rapid, but deteriorates as smooth error components, those with large eigenvalues, dominate the remaining error. The purpose of transferring the problem to a coarser grid is to make these smooth error components appear more oscillatory with respect to the grid spacing, so that the initial rapid convergence rate is obtained for the elimination of these smooth errors by coarse-grid iterations. Since the coarse grid Q2h has only 1/4 as many grid points as Qh (in 2-d), the smoothing iterations on the coarse grid are cheaper as well as more effective in reducing the smooth error components than on the fine grid. In the correction scheme, the coarse-grid problem is an equation for the algebraic error, A2he2h ^ r2h, (4.3) approximating the fine-grid residual equation for the algebraic error. To obtain the coarse-grid source term, r2h, the restriction procedure Ih is applied to the fine-grid residual rh, r2h I2hh. (4.4) Eq. 4.4 is an averaging type of operation. Two common restriction procedures are straight injection of fine-grid values to their corresponding coarse-grid grid points, and averaging rh over a few fine-grid grid points which are near the corresponding coarse-grid grid point. The initial error on the coarse grid is taken as zero. After the solution for e2h is obtained, this coarse-grid quantity is interpolated to the fine grid and used to correct the fine-grid solution, ^ + 2h. (4.5) For Ihh, common choices are bilinear or biquadratic interpolation. In practice the solution for e2h is obtained by recursion on the two-level cycle- (A2h)-1 is not explicitly computed. On the coarsest grid, direct solution may be feasible if the equation is simple enough. Otherwise a few smoothing iterations can be applied. Recursion on the two-level algorithm leads to a "V cycle," as shown in Figure 4.1. A simple V(3,2) cycle is shown. Three smoothing iterations are taken before re- stricting to the next coarser grid, and two iterations are taken after the solution has been corrected. The purpose of the latter smoothing iterations is to smooth out any high-frequency noise introduced by the prolongation. Other cycles can be envi- sioned. In particular the W cycle is popular [6]. The cycling strategy is called the "grid-schedule," since it is the order in which the various grid levels are visited. The most important consideration for the correction scheme has been saved for last, namely the definition of the coarse-grid discrete equation A2h. One possibility is to discretize the original differential equation directly on the coarse grid. However this choice is not always the best one. The convergence-rate benefit from the multigrid strategy is derived from the particular coarse-grid approximation to the fine-grid discrete problem, not the continuous problem. Because the coarse-grid solutions and residuals are obtained by particular averaging procedures, there is an implied averaging procedure for the fine-grid discrete operator Ah which should be honored to ensure a useful homogenization of the fine-grid residual equation. This issue is critical when the coefficients and/or dependent variables of the governing equations are not smooth [17]. For the Poisson equation, the Galerkin approximation A2h 1= 2hIA^I2h is the right choice. The discretized equation coefficients on the coarse grid are obtained by applying suitable averaging and interpolation operations to the fine-grid coeffi- cients, instead of by discretizing the governing equation on a grid with a coarser mesh spacing. Briggs has shown, by exploiting the algebraic relationship between bilinear interpolation and full-weighting restriction operators, that initially smooth errors begin in the range of interpolation and finish, after the smoothing-correction cycle is applied, in the null space of the restriction operator [10]. Thus, if the fine-grid smoothing eliminates all the high-frequency error components in the solution, one V cycle using the correction-scheme is a direct solver for the Poisson equation. The con- vergence rate of multigrid methods using the Galerkin approximation is more difficult to analyze if the governing equations are more complicated than Poisson equations, but significant theoretical advantages for application to general linear problems have been indicated [90]. 4.1.2 Full-Approximation Storage Scheme for Nonlinear Equations The brief description given above does not bring out the complexities inherent in the application to nonlinear problems. There is only experience, derived mostly from numerical experiments, to guide the choice of the restriction/prolongation procedures and the smoother. Furthermore, the linkage between the grid levels requires special considerations because of the nonlinearity. The correction scheme using the Galerkin approximation can be applied to the nonlinear Navier-Stokes system of equations [94]. However, in order to use CS for nonlinear equations, linearization is required. The best coarse-grid correction only improves the fine-grid solution to the linearized equation. Also, for complex equa- tions, considerable expense is incurred in computing A2h by the Galerkin approxi- mation. The commonly adopted alternative is the intuitive one, to let A2h be the differential operator L discretized on the grid with spacing 2h instead of h. In ex- change for a straightforward problem definition on the coarse grid though, special restriction and prolongation procedures may be necessary to ensure the usefulness of the resulting corrections. Numerical experiments on a problem-by-problem basis are necessary to determine good choices for the restriction and prolongation procedures for Navier-Stokes multigrid methods. The full-approximation storage (FAS) scheme [5] is preferred over the correction scheme for nonlinear problems. The coarse-grid corrections generated by FAS improve the solution to the full nonlinear problem instead of just the linearized one. The discretized equation on the fine grid is, again, Ahuh = Sh. (4.6) The approximate solution vh after a few fine-grid iterations defines the residual on the fine grid, Ahvh = Sh + rh. (4.7) A correction, the algebraic error e^l = uh vh, is sought which satisfies Ah(vh + el) = Sh. (4.8) The residual equation is formed by subtracting Eq. 4.7 from Eq. 4.8, and cancelling Sh, Ah(vh + eh) Ah(vh) = -rh, (4.9) where the subscript "alg" is dropped for convenience. For linear equations the Ahvh terms cancel leaving Eq. 4.3. Eq. 4.9 does not simplify for nonlinear equations. Assuming that the smoother has done its job, rh is smooth and Eq. 4.9 is the same as the coarse-grid residual equation A2h(2h + e2h) A2h(2h) = -r2h, (4.10) at coarse-grid grid points. The error e2h is to be found, interpolated back to ih according to eh = Ih^2h, and added to vh so that Eq. 4.8 is satisfied. The known quantities are v2h, which is a "suitable" restriction of vh, and r2h, likewise a restriction of rh. Different restrictions can be used for residuals and solutions. Thus, Eq. 4.10 can be written A2h(Ihvh + 62h) = A2h(Ir hv) I7^rh. (4.11) Since Eq. 4.11 is not an equation for e2h, one solves instead for the sum Ihhvh + e2h Expanding rh and regrouping terms, Eq. 4.11 can be written A2h(u2h) = A2h(I hvh) I2h h (4.12) = [A2h(Ihvh) Ih2h(Ahvh) + IhhSh S2h] + S2h (4.13) [Snumerical 2+ S2h, (4.14) Eq. 4.14 is similar to Eq. 4.6 except for the extra numerically-derived source term. Once I^hvh + e2h is obtained the coarse-grid approximation to the fine-grid error, e2h, is computed by first subtracting the initial coarse-grid solution I^hvh, e2h = 2h I2h, (4.15) then interpolating back to the fine grid and combining with the current solution, vh + vh + I2h(e2h). (4.16) 4.1.3 Extension to the Navier-Stokes Equations The incompressible Navier-Stokes equations are a system of coupled, nonlinear equations. Consequently the FAS scheme given above for single nonlinear equations needs to be modified. The variables u^, u^, and uh represent the cartesian velocity components and the pressure, respectively. Corresponding subscripts are used to identify each equations' source term, residual and discrete operator in the formulation below. The three equations for momentum and mass conservation are treated as if part of the following matrix equation, A 0 Gh uh SS^ 0 Ah G [ U^ S2^ (4.17) Gh G h 0 U S^h The continuity equation source term is zero on the finest grid, Qh, but for coarser grid levels it may not be zero. Thus, for the sake of generality it is included in Eq. 4.17. Thus, for the ux-momentum equation Eq. 4.8 is modified to account for the pressure-gradient, G^u^, which is also an unknown. The approximate solutions are v^, vh, and v3 corresponding to u^, u4, and u.. For the ul-momentum equation, the approximate solution satisfies Av + G^v^ = S^ + ,^. (4.18) The fine-grid residual equation corresponding to Eq. 4.9 is modified to Al(v1 + eh) Ah(v ) + G (v3 + eh) G(h) = -rh, (4.19) which is approximated on the coarse grid by the corresponding coarse-grid residual equation, A2h(vh + e2h) A2h(vh) + Gh(v + e^) G2v) = (4.20) 1 1V -11 x M + e3) Mv .r The known terms are v2h = Ihh, v2h h= hh, and r2h = I2h ^ Expanding r and regrouping terms, Eq. 4.19 can be written Expanding rh and regrouping terms, Eq. 4.19 can be written Ah(uh) + Gh (u2h) A2hr h2h h 2h 2h h = Al (f, ) + Gx (l v^3) -2h (A^hvh + GGVh) + I2hSh 2hl i2h h) 2h r2h h = [A (h + Gx (1h v 3) I2h Ahv + G)+ Ih2hSh Sh] + s2h -h ( IU + Gv1 3 1 1 1h 1Ll,nulmer icalrc I * Since Eq. 4.22 includes numerically derived source terms in addition to the physical ones, the coarse-grid variables are not in general the same as would be obtained from a discretization of the original continuous governing equations on the coarse grid. The u2-momentum equation is treated similarly, and the coarse-grid continuity equation is G2hu2h + G2hU2h = G ^2h(I h h\ G2h(lr2hi h\ 12h r X1 y 2 x h UI y 2h 'h 3 (4.22) (4.21) |

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BLOSCH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 ACKNOWLEDGEMENTS I would like to express my thanks to my advisor Dr. Wei Shyy for reflecting carefully on my results and for directing my research toward interesting issues. I would also like to thank him for the exceptional personal support and flexibility he offered me during my last year of study, which was done off-campus. I would also like to acknowledge the contributions of the other members of my Ph.D. committee, Dr. Chen-Chi Hsu, Dr. Bruce Carroll, Dr. David Mikolaitis, and Dr. Sartaj Sahni. Dr. Hsu and Dr. Carroll supervised my B.S. and M.S. degree research studies, respectively, and Dr. Mikolaitis, in the role of graduate coordinator, enabled me to obtain financial support from the Department of Energy. Also I would like to thank Madhukar Rao, Rick Smith and H.S. Udaykumar, for paying fees on mv behalf and for registering me for classes while I was in California. Jeff Wright, S. Thakur, Shin-Jye Liang, Guobao Guo and Pedro Lopez-Fernandez have also made direct and indirect contributions for which I am grateful. Special thanks go to Dr. Jamie Sethian, Dr. Alexandre Chorin and Dr. Paul Con- cus of Lawrence Berkeley Laboratory for allowing me to visit LBL and use their resources, for giving personal words of support and constructive advice, and for the privilege of interacting with them and their graduate students in the applied matheÂ¬ matics branch. Last but not least I would like to thank my wife, Laura, for her patience, her example, and her frank thoughts on â€œcups with sliding lids,â€ â€œflow through straws,â€ and numerical simulations in general. n My research was supported in part by the Computational Science Graduate FelÂ¬ lowship Program of the Office of Scientific Computing in the Department of Energy. The CM-5s used in this study were partially funded by National Science Foundation Infrastructure Grant CDA-8722788 (in the computer science department of the UniÂ¬ versity of California-Berkeley), and a grant of HPC time from the DoD HPC Shared Resource Center, Army High-Performance Computing Research Center, Minneapolis, Minnesota. m TABLE OF CONTENTS ACKNOWLEDGEMENTS ii ABSTRACT vi CHAPTERS 1 INTRODUCTION 1 1.1 Motivations 1 1.2 Governing Equations 3 1.3 Numerical Methods for Viscous Incompressible Flow 5 1.4 Parallel Computing 7 1.4.1 Data-Parallelism and SIMD Computers 8 1.4.2 Algorithms and Performance 11 1.5 Pres sure-Based Multigrid Methods 13 1.6 Description of the Research 17 2 PRESSURE-CORRECTION METHODS 21 2.1 Finite-Volume Discretization on Staggered Grids 21 2.2 The SIMPLE Method _ 23 2.3 Discrete Formulation of the Pressure-Correction Equation 27 2.4 Well-Posedness of the Pressure-Correction Equation 30 2.4.1 Analysis 30 2.4.2 Verification by Numerical Experiments 33 2.5 Numerical Treatment of Outflow Boundaries 38 2.6 Concluding Remarks 40 3 EFFICIENCY AND SCALABILITY ON SIMD COMPUTERS 53 3.1 Background 53 3.1.1 Speedup and Efficiency 53 3.1.2 Comparison Between CM-2, CM-5, and MP-1 55 3.1.3 Hierarchical and Cut-and-Stack Data Mappings 57 3.2 Implementional Considerations 59 3.3 Numerical Experiments 61 3.3.1 Efficiency of Point and Line Solvers for the Inner Iterations . . 62 3.3.2 Effect of Uniform Boundary Condition Implementation .... 69 3.3.3 Overall Performance 70 3.3.4 Isoefficiency Plot 72 IV 3.4 Concluding Remarks 74 4 A NONLINEAR PRESSURE-CORRECTION MULTIGRID METHOD . . 83 4.1 Background 84 4.1.1 Terminology and Scheme for Linear Equations 86 4.1.2 Full-Approximation Storage Scheme for Nonlinear Equations . 90 4.1.3 Extension to the Navier-Stokes Equations 92 4.2 Comparison of Pressure-Based Smoothers 94 4.3 Stability of Multigrid Iterations 101 4.3.1 Defect-Correction Method 103 4.3.2 Cost of Different Convection Schemes 106 4.4 Restriction and Prolongation Procedures 108 4.5 Concluding Remarks 113 5 IMPLEMENTATION AND PERFORMANCE ON THE CM-5 127 5.1 Storage Problem 128 5.2 Multigrid Convergence Rate and Stability 131 5.2.1 Truncation Error Convergence Criterion for Coarse Grids ... 133 5.2.2 Numerical Characteristics of the FMG Procedure 136 5.2.3 Influence of Initial Guess on Convergence Rate 145 5.2.4 Remarks 148 5.3 Performance on the CM-5 149 5.4 Concluding Remarks 156 REFERENCES 181 BIOGRAPHICAL SKETCH 188 v Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRESSURE-BASED METHODS ON SINGLE-INSTRUCTION STREAM/MULTIPLE-DATA STREAM COMPUTERS By Edwin L. Blosch Chairman: Dr. Wei Shyy Major Department: Aerospace Engineering, Mechanics and Engineering Science Computationally and numerically scalable algorithms are needed to exploit emergÂ¬ ing parallel-computing capabilities. In this work pressure-based algorithms which solve the two-dimensional incompressible Navier-Stokes equations are developed for single-instruction stream/multiple-data stream (SIMD) computers. The implications of the continuity constraint for the proper numerical treatment of open boundary problems are investigated. Mass must be conserved globally so that the system of linear algebraic pressure-correction equations is numerically consistent. The convergence rate is poor unless global mass conservation is enforced explicitly. Using an additive-correction technique to restore global mass conservation, flows which have recirculating zones across the open boundary can be simulated. The performance of the single-grid algorithm is assessed on three massively- parallel computers, MasParâ€™s MP-1 and Thinking Machinesâ€™ CM-2 and CM-5. ParalÂ¬ lel efficiencies approaching 0.8 are possible with speeds exceeding that of traditional vector supercomputers. The following issues relevant to the variation of parallel efÂ¬ ficiency with problem size are studied: the suitability of the algorithm for SIMD computation; the implementation of boundary conditions to avoid idle processors; vi the choice of point versus line-iterative relaxation schemes; the relative costs of the coefficient computations and solving operations, and the variation of these costs with problem size; the effect of the data-array-to-processor mapping; and the relative speeds of computation and communication of the computer. A nonlinear pressure-correction multigrid algorithm which has better convergence rate characteristics than the single-grid method is formulated and implemented on the CM-5. On the CM-5, the components of the multigrid algorithm are tested over a range of problem sizes. The smoothing step is the dominant cost. Pressure-correction methods and the locally-coupled explicit method are equally efficient on the CM-5. V cycling is found to be much cheaper than W cycling, and a truncation-error based â€œfull-multigridâ€ procedure is found to be a computationally efficient and convenient method for obtaining the initial fine-grid guess. The findings presented enable further development of efficient, scalable pressure-based parallel computing algorithms. Vll CHAPTER 1 INTRODUCTION 1.1 Motivations Computational fluid dynamics (CFD) is a growing field which brings together high-performance computing, physical science, and engineering technology. The disÂ¬ tinctions between CFD and other fields such as computational physics and computaÂ¬ tional chemistry are largely semantic now, because increasingly more interdisplinary applications are coming within range of the computational capabilities. CFD algoÂ¬ rithms and techniques are mature enough that the focus of research is expected to shift in the next decade toward the development of robust flow codes, and toward the application of these codes to numerical simulations which do not idealize either the physics or the geometry and which take full account of the coupling between fluid dynamics and other areas of physics [65]. These applications will require formidable resources, particularly in the areas of computing speed, memory, storage, and inÂ¬ put/output bandwidth [78]. At the present time, the computational demands of the applications are still at least two orders-of-magnitude beyond the computing technology. For example, NASAâ€™s grand challenges for the 1990s are to achieve the capability to simulate visÂ¬ cous, compressible flows with two-equation turbulence modelling over entire aircraft configurations, and to couple the fluid dynamics simulation with the propulsion and aircraft control systems modelling. To meet this challenge it is estimated that 1 ter- aflops computing speed and 50 gigawords of memory will be required [24]. Current 1 2 massively-parallel supercomputers, for example, the CM-5 manufactured by Thinking Machines, have peak speeds of 0(10 gigaflops) and memories of 0(1 gigaword). Optimism is sometimes circulated that teraflop computers may be expected by 1995 [68]. In view of the two orders-of-magnitude disparity between the speed of present-generation parallel computers and teraflops, such optimism should be dimmed somewhat. Expectations are not being met in part because the applications, which are the driving force behind the progress in hardware, have been slow to develop. The numerical algorithms which have seen two decades of development on traditional vecÂ¬ tor supercomputers are not always easy targets for efficient parallel implementation. Better understanding of the basic concepts and more experience with the present generation of parallel computers is a prerequisite for improved algorithms and impleÂ¬ mentations. The motivation of the present work has been the opportunity to investigate issues related to the use of parallel computers in CFD, with the hope that the knowledge gained can assist the transition to the new computing technology. The context of the research is the numerical solution of the 2-d incompressible Navier-Stokes equations, by a popular and proven numerical method known as the pressure-correction techÂ¬ nique. A specific objective emerged as the research progressed, namely to develop and analyze the performance of pressure-correction methods on the single-instruction stream/multiple-data stream (SIMD) type of parallel computer. Single-grid compuÂ¬ tations were studied first, then a multigrid method was developed and tested. StMD computers were chosen because they are easier to program than multiple- instruction stream/multiple-data stream (MIMD) computers (explict message-passing is not required), because synchronization of the processors is not an issue, and beÂ¬ cause the factors affecting the parallel run time and computational efficiency are easier to identify and quantify. Also, these are arguably the most powerful machines 3 available right nowâ€”Los Alamos National Laboratory has a 1024-node CM-5 with 32 Gbytes of processor memory and is capable of 32 Gflops peak speed. Thus, the code, the numerical techniques, and the understanding which are the contribution of this research can be immediately useful for applications on massively parallel computers. 1.2 Governing Equations The governing equations for 2-d, constant property, time-dependent viscous inÂ¬ compressible flow are the Navier-Stokes equations. They express the principles of conservation of mass and momentum. In primitive variables and cartesian coordiÂ¬ nates, they may be written dpu dpv dx dy dpu dpu2 dpuv dp d2u d2u ~df + ~d^ + ~d^~ = ~Tx + ^d^ + dpv dpuv dpv2 dp d2v d2v ^w + ~d^ + ~d^~ = ~d^ + fld^ + ^d^ (1.1) (1.2) (1.3) where u and v are cartesian velocity components, p is the density, p is the fluidâ€™s molecular viscosity, and p is the pressure. Eq. 1.1 is the mass continuity equation, also known as the divergence-free constraint since its coordinate-free form is div u = 0. The Navier-Stokes equations 1.1-1.3 are a coupled set of nonlinear partial differÂ¬ ential equations of mixed elliptic/parabolic type. Mathematically, they differ from the compressible Navier-Stokes equations in two important respects that lead to difÂ¬ ficulties for devising numerical solution techniques. First, the role of the continuity equation is different in incompressible flow. InÂ¬ stead of a time-dependent equation for the density, in incompressible fluids the contiÂ¬ nuity equation is a constraint on the admissible velocity solutions. Numerical methÂ¬ ods must be able to integrate the momentum equations forward in time while simulÂ¬ taneously maintaining satisfaction of the continuity constraint. On the other hand, 4 numerical methods for compressible flows can take advantage of the fact that in the unsteady form each equation has a time-dependent term. The equations are cast in vector formâ€”any suitable method for time-integration can be employed on the system of equations as a whole. The second problem, assuming that a primitive-variable formulation is desired, is that there is no equation for pressure. For compressible flows, the pressure can be deÂ¬ termined from the equation of state of the fluid. For incompressible flow, an auxiliary â€œpressure-Poissonâ€ equation can be derived by taking the divergence of the vector form of the momentum equations; the continuity equation is invoked to eliminate the unsteady term in the result. The formulation of the pressure-Poisson equation requires manipulating the discrete forms of the momentum and continuity equations. A particular discretization of the Laplacian operator is therefore implied in pressure- Poisson equation, depending on the discrete gradient and divergence operators. This operator may not be implementable at boundaries, and solvability constraints can be violated [30]. Also, the differentiation of the governing equations introduces the need for additional unphysical boundary conditions on the pressure. Physically, the pressure in incompressible flow is only defined relative to an (arbitrary) constant. Thus, the correct boundary conditions are Neumann. However, if the problem has an open boundary, the governing equations should be supplemented with a boundary condition on the normal traction [29, 32], Fn = â€”p + 1 dun Re dn (1.4) where F is the force, Re is the Reynolds number, and the subscript n indicates the normal direction. However, Fn may be difficult to prescribe. 5 In practice, a zero-gradient or linear extrapolation for the normal velocity comÂ¬ ponent is a more popular outflow boundary condition. Many outflow boundary conÂ¬ ditions have been analyzed theoretically for incompressible flow (see [30, 31, 38, 56]). There are even more boundary condition procedures in use. The method used and its impact on the â€œsolvabilityâ€™' of the resulting numerical systems of equations depends on the discretization and the numerical method. This issue is treated in Chapter 2. 1.3 Numerical Methods for Viscous Incompressible Flow Numerical algorithms for solving the incompressible Navier-Stokes system of equaÂ¬ tions were first developed by Harlow and Welch [39] and Chorin [15, 16]. Descendants of these approaches are popular today. Harlow and Welch introduced the important contribution of the staggered-grid location of the dependent variables. On a stagÂ¬ gered grid, the discrete Laplacian appearing in the derivation of the pressure-Poisson equation has the standard five-point stencil. On colocated grids it still has a five- point form but, if the central point is located at (i,j), the other points which are involved are located at (i+2,j), (i-2,j), (i,j+2), and (i,j-2). Without nearest-neighbor linkages, two uncoupled (â€œcheckerboardâ€) pressure fields can develop independently. This pressure-decoupling can cause stability problems, since nonphysical discontinuÂ¬ ities in the pressure may develop [50]. In the present work, the velocity components are staggered one-half of a control volume to the west and south of the pressure which is defined at the center of the control volume as shown in Figure 1.1. Figure 1.1 also shows the locations of all boundary velocity components involved in the discretization and numerical solution, and representative boundary control volumes for u, v, and p. In Chorinâ€™s artificial compressibility approach [15] a time-derivative of pressure is added to the continuity equation. In this manner the continuity equation becomes an equation for the pressure, and all the equations can be integrated forward in time, 6 either as a system or one at a time. The artificial compressibility method is closely related to the penalty formulation used in finite-element methods [41]. The equations are solved simultaneously in finite-element formulations. Penalty methods and the artificial compressibility approach suffer from ill-conditioning when the equations have strong nonlinearities or source terms. Because the pressure term is artificial, they are not time-accurate either. Projection methods [16, 62] are two-step procedures which first obtain a velocity field by integrating the momentum equations, and then project this vector field into a divergence-free space by subtracting the gradient of the pressure. The pressure- Poisson equation is solved to obtain the pressure. The solution must be obtained to a high degree of accuracy in unsteady calculations in order to obtain the correct long-term behavior [76]â€”every step may therefore be fairly expensive. Furthermore, the time-step size is limited by stability considerations, depending on the implicitness of the treatment used for the convection terms. â€œPressure-basedâ€ methods for the incompressible Navier-Stokes equations include SIMPLE [61] and its variants, SIMPLEC [19], SIMPLER [60], and PISO [43]. These methods are similar to projection methods in the sense that a non-mass-conserving velocity field is computed first, and then corrected to satisfy continuity. However, they are not implicit in two steps because the nonlinear convection terms are linearized explicitly. Instead of a pressure-Poisson equation, an approximate equation for the pressure or pressure-correction is derived by manipulating the discrete forms of the momentum and continuity equations. A few iterations of a suitable relaxation method are used to obtain a partial solution to the system of correction equations, and then new guesses for pressure and velocity are obtained by adding the corrections to the old values. This process is iterated until all three equations are satisfied. The iterations require underrelaxation because of the sequential coupling between 7 variables. Compared to projection methods, pressure-based methods are less implicit when used for time-dependent problems. However, they can be used to seek the steady-state directly if desired. Compared to a fully coupled strategy, the sequential pressure-based approach typically has slower convergence and less robustness with respect to Reynolds numÂ¬ ber. However, the sequential approach has the important advantage that additional complexities, for example, chemical reaction, can be easily accommodated by simply adding species-balance equations to the stack. The overall run time increases since each governing equation is solved independently, and the total storage requirements scale linearly with the number of equations solved. On the other hand, the computer time and storage requirements escalate faster in a fully coupled solution strategy. The typical way around this problem is to solve simultaneously the continuity and momenÂ¬ tum equations, then solve any additional equations in a sequential fashion. Without knowing beforehand that the pressure-velocity coupling is the strongest among all the various flow variables, however, the extra computational effort spent in simultaneous solution of these equations is unwarranted. There are other approaches for solving the incompressible Navier-Stokes equaÂ¬ tions, notably methods based on vorticity-streamfunction â€” or velocity-vorticity (u â€” u) formulations, but pressure-based methods are easier, especially with regard to boundary conditions and possible extension to 3-d domains. Furthermore, they have demonstrated considerable robustness in computing incompressible flows. A broad range of applications of pressure-based methods is demonstrated in [73]. 1.4 Parallel Computing General background of parallel computers and their application to the numeriÂ¬ cal solution of partial differential equations is given in Hockney and Jesshope [40] 8 and Ortega and Voigt [58]. Fischer and Patera [23] gave a recent review of parallel computing from the perspective of the fluid dynamics community. Their â€œindirect cost,â€ the parallel run time, is of primary interest here. The â€œdirect costâ€ of parallel computers and their components is another matter entirely. For the iteration-based numerical methods developed here, the parallel run time is the cost per iteration multiplied by the number of iterations. The latter is affected by the characteristics of the particular parallel computer used and the algorithms and implementations emÂ¬ ployed. Parallel computers come in all shapes and sizes, and it is becoming virtually impossible to give a thorough taxonomy. The background given here is limited to a description of the type of computer used in this work. 1.4.1 Data-Parallelism and SIMP Computers Single-instruction stream/multiple-data stream (SIMD) computers include the connection machines manufactured by the Thinking Machines Corporation, the CM and CM-2, and the MP-1, MP-2, and MP-3 computers produced by the MasPar CorÂ¬ poration. These are massively-parallel machines consisting of a front-end computer and many processor/memory pairs, figuratively, the â€œback-end.â€ The back-end proÂ¬ cessors are connected to each other by a â€œdata network.â€ The topology of the data network is a major feature of distributed-memory parallel computers. The schematic in Figure 1.2 gives the general idea of the SIMD layout. The program executes on the serial front-end computer. The front-end triggers the synÂ¬ chronous execution of the â€œback-endâ€ processors by sending â€œcode blocksâ€ simulÂ¬ taneously to all processors. Actually, the code blocks are sent to an intermediate â€œcontrol processor(s).â€ The control processor broadcasts the instructions contained 9 in the code block, one at a time, to the computing processors. These â€œfront-end- to-processorâ€ communications take time. This time is an overhead cost not present when the program runs on a serial computer. The operands of the instructions, the data, are distributed among the processorsâ€™ memories. Each processor operates on its own locally-stored data. The â€œdataâ€ in grid-based numerical methods are the arrays, 2-d in this case, of dependent variables, geometric quantities, and equation coefficients. Because there are usually plenty of grid points and the same governing equations apply at each point, most CFD algorithms contain many operations to be performed at every grid point. Thus this â€œdata-parallelâ€ approach is very natural to most CFD algorithms. Many operations may be done independently on each grid point, but there is couÂ¬ pling between grid points in physically-derived problems. The data network enters the picture when an instruction involves another processorâ€™s data. Such â€œinterproÂ¬ cessorâ€ communication is another overhead cost of solving the problem on a parallel computer. For a given algorithm, the amount of interprocessor communication deÂ¬ pends on the â€œdata mapping,â€ which refers to the partitioning of the arrays and the assignment of these â€œsubgridsâ€ to processors. For a given machine, the speed of the interprocessor communication depends on the pattern of communication (random or regular) and the distance between the processors (far away or nearest-neighbor). The run time of a parallel program depends first on the amount of front-end and parallel computation in the algorithm, and the speeds of the front-end and backÂ¬ end for doing these computations. In the programs developed here, the front-end computations are mainly the program control statements (IF blocks, DO loops, etc.). The front-end work is not sped up by parallel processing. The parallel computations are the useful work, and by design one hopes to have enough parallel computation 10 to amortize both the front-end computation and the interprocessor and front-end-to- processor communication, which are the other factors that contribute to the parallel run time. From this brief description it should be clear that SIMD computers have four charÂ¬ acteristic speeds: the computation speed of the processors, the communication speed between processors, and the speed of the front-end-to-processor communication, i.e. the speed that code blocks are transferred, and the speed of the front-end. These machine characteristics are not under the control of the programmer. However, the amount of computation and communication a program contains is determined by the programmer because it depends on the algorithm selected and the algorithmâ€™s impleÂ¬ mentation (the choice of the data mapping, for example). Thus, the key to obtaining good performance from SIMD computers is to pick a suitable algorithm, â€œmatchedâ€ in a sense to the architecture, and to develop an implementation which minimizes and localizes the interprocessor communication. Then, if there is enough parallel computation to amortize the serial content of the program and the communication overheads, the speedup obtained will be nearly the number of processors. The actual performance, because it depends on the computer, the algorithm, and the impleÂ¬ mentation, must be determined by numerical experiment on a program-by-program basis. SIMD computers are restricted to exploiting data-parallelism, as opposed to the parallelism of the tasks in an algorithm. The task-parallel approach is more comÂ¬ monly used, for example, on the Cray C90 supercomputer. Multiple-instruction stream/multiple-data stream (MIMD) computers, on the other hand, are composed of more-or-less autonomous processor/memory pairs. Examples include the Intel series of machines (iPSC/2, iPSC/860, and Paragon), workstation clusters, and the connecÂ¬ tion machine CM-5. However, in CFD, the data-parallel approach is the prevalent 11 one even on MIMD computers. The front-end/back-end programming paradigm is implemented by selecting one processor to initiate programs on the other processors, accumulate global results, and enforce synchronization when necessary, a strategy called single-program-multiple-data (SPMD) [23]. The CM-5 has a special â€œcontrol networkâ€ to provide automatic synchronization of the processorâ€™s execution, so a SIMD programming model can be supported as well as MIMD. SIMD is the manner in which the CM-5 has been used in the present work. The advantage to using the CM-5 in the SIMD mode is that the programmer does not have to explicitly specify message-passing. This simplification saves effort and increases the effective speed of communication because certain time-consuming protocols for the data transfer can be eliminated. 1.4.2 Algorithms and Performance The previous subsection discussed data-parallelism and SIMD computers, i.e. what parallel computing means in the present context and how it is carried out by SIMD-tvpe computers. To develop programs for SIMD computers requires one to recognize that unlike serial computers, parallel computers are not black boxes. In addition to the selection of an algorithm with ample data-parallelism, consideration must be given to the implementation of the algorithm in specific ways in order to achieve the desired benefits (speedups over serial computations). The success of the choice of algorithm and the implementation on a particular computer is judged by the â€œspeedupâ€ (S) and â€œefficiencyâ€ (E) of the program. The communications mentioned above, front-end-to-processor and interprocessor, are esÂ¬ sentially overhead costs associated with the SIMD computational model. They would not be present if the algorithm were implemented on a serial computer, or if such communications were infinitely fast. If the overhead cost was zero, a parallel program 12 executing on np processors would run np times faster than on a single processor, a speedup of np. This idealized case would also have a parallel efficiency of 1. The parallel efficiency E measures the actual speedup in comparison with the ideal. One is also interested in how speedup, efficiency, and the parallel run time (Tp) scale with problem size, and with the number of processors used. The objective in using parallel computers is more than just obtaining a good speedup on a particular problem size and a particular number of processors. For parallel CFD, the goals are to either (1) reduce the time (the indirect cost [23]) to solve problems of a given complexity, to satisfy the need for rapid turnaround times in design work, or (2) increase the complexity of problems which can be solved in a fixed amount of time. For the iteration-based numerical methods studied here, there are two considerations: the cost per iteration, and the number of iterations, respectively, computational and numerical factors. The total run time is the product of the two. Gustafson [35] has presented fixed-size and scaled-size experiments whose results describe how the cost per iteration scales on a particular machine. In the fixed- size experiment, the efficiency is measured for a fixed problem size as processors are added. The hope is that the run time is halved when the number of processors is doubled. However, the run time obviously cannot be reduced indefinitely by adding more processors because at some point the parallelism runs outâ€”the limit to the attainable speedup is the number of grid points. In the scaled-size experiment, the problem size is increased along with the number of processors, to maintain a constant local problem size for each of the parallel processors. Care must be taken to make timings on a per iteration basis if the number of iterations to reach the end of the computation increases with the problem size. The hope in such an experiment is that the program will maintain a certain high level of parallel efficiency E. The ability 13 to maintain E in the scaled-size experiment indicates that the additional processors increased the speedup in a one-for-one trade. 1.5 Pressure-Based Multigrid Methods Multigrid methods are a potential route to both computationally and numerically scalable programs. Their cost per iteration on parallel computers and convergence rate is the subject of Chapters 4-5. For sufficiently smooth elliptic problems, the convergence rate of multigrid methods is independent of the problem sizeâ€”their opÂ¬ eration count is 0(N). In practice, good convergence rates are maintained as the problem size increases for Navier-Stokes problems, also, provided suitable multigrid componentsâ€”the smoother, restriction and prolongation proceduresâ€”and multigrid techniques are employed. The standard V-cycle full-multigrid (FMG) algorithm has an almost optimal operation count, 0(log2N) for Poisson equations, on parallel comÂ¬ puters. Provided the multigrid algorithm is implemented efficiently and that the cost per iteration scales well with the problem size and the number of processors, the multigrid approach seems to be a promising way to exploit the increased computaÂ¬ tional capabilities that parallel computers offer. The pressure-based methods mentioned previously involve the solution of three systems of linear algebraic equations, one each for the two velocity components and one for the pressure, by standard iterative methods such as successive line- underrelaxation (SLUR). Hence they inherit the convergence rate properties of these solvers, i.e. as the problem size grows the convergence rate deteriorates. With the single-grid techniques, therefore, it will be difficult to obtain reasonable turnaround times when the problem size is increased into the target range for parallel comÂ¬ puters. Multigrid techniques for accelerating the convergence of pressure-correction 14 methods should be pursued, and in fact they have been within the last five or so years [70, 74, 80]. However, there are still many unsettled issues. The complexities affecting the convergence rate of single-grid calculations carry over to the multigrid framework and are compounded there by the coupling between the evolving solutions on multiple grid levels, and by the particular â€œgrid-schedulingâ€ used. Linear multigrid methods have been applied to accelerate the convergence rate for the solution of the system of pressure or pressure-correction equations [4, 22, 42, 64, 94], However, the overall convergence rate does not significantly improve because the velocity-pressure coupling is not addressed [4, 22]. Therefore the multigrid strategy should be applied on the â€œouter loop,â€ with the role of the iterative relaxation method played by the numerical methods described above, e.g. the projection method or the pressure-correction method. Thus, the generic term â€œsmootherâ€ is prescribed because it reflects the purpose of the solution of the coupled system of equations going on inside the multigrid cycleâ€”to smooth the residual so that an accurate coarse-grid approximation of the fine-grid problem is possible. It is not true that a good solver, one with a fast convergence rate on single-grid computations, is necessarily a good smoother of the residual. It is therefore of interest to assess pressure-correction methÂ¬ ods as potential multigrid smoothers. See Shyy and Sun [74] for more information on the staggered-grid implementation of multigrid methods, and some encouraging results. Staggered grids require special techniques [21, 74] for the transfer of solutions and residuals between grid levels, since the positions of the variables on different levels do not correspond. However, they alleviate the â€œcheckerboardâ€ pressure stability problem [50], and since techniques have already been established [74], there is no 15 reason not to go this route, especially when cartesian grids are used as in the present work. Vanka [89] has proposed a new numerical method as a smoother for multigrid computations, one which has inferior convergence properties as a single-grid method but apparently yields an effective multigrid method. A staggered-grid finite-volume discretization is employed. In Vankaâ€™s smoother, the velocity components and presÂ¬ sure of each control volume are updated simultaneously, so it is a coupled approach, but the coupling between control volumes is not taken into account, so the calcuÂ¬ lation of new velocities and pressures is explicit. This method is sometimes called the â€œlocally-coupled explicitâ€ or â€œblock-explicitâ€ pressure-based method. The control volumes are visited in lexicographic order in the original method which is therefore aptly called BGS (block Gauss-Seidel). Line-variants have been developed to couple the flow variables in neighboring control volumes along lines (see [80, 87]). Linden et al. [50] gave a brief survey of multigrid methods for the steady-state inÂ¬ compressible Navier-Stokes equations. They argue without analysis that BGS should be preferred over the pressure-correction type methods since the strong local couÂ¬ pling is likely to have better success smoothing the residual locally. On the other hand, Sivaloganathan and Shaw [71, 70] have found good smoothing properties for the pressure-correction approach, although the analysis was simplified considerably. Sockol [80] has compared the point and line-variants of BGS with the pressure- correction methods on serial computers, using model problems with different physical characteristics. SIMPLE and BGS emerge as favorites in terms of robustness with BGS preferred due to a lower cost per iteration. This preference may or may not carry over to SIMD parallel computers (see Chapter 4 for comparison). Interesting applications of multigrid methods to incompressible Navier-Stokes flow problems can be found in [12, 28, 48, 54]. 16 In terms of parallel implementations there are far fewer results although this field is rapidly growing. Simon [77] gives a recent cross-section of parallel CFD results. Parallel multigrid methods, not only in CFD but as a general technique for partial differential equations, have received much attention due to their desirable 0(N) operation count on Poisson equations. However, it is apparently difficult to find or design parallel computers with ideal communication networks for multigrid [13]. Consequently implementations have been pursued on a variety of machines to see what performance can be obtained with the present generation of parallel machines, and to identify and understand the basic issues. Dendy et al.[18] have recently described a multigrid method on the CM-2. However, to accommodate the data- parallel programming model they had to dimension their array data on every grid level to the dimension extents of the finest grid array data. This approach is very wasteful of storage. Consequently the size of problems which can be solved is greatly reduced. Recently an improved release of the compiler has enabled the storage problem to be circumvented with some programming diligence (see Chapter 5). The implementation developed in this work is one of the first to take advantage of the new compiler feature. In addition to parallel implementations of serial multigrid algorithms, several novel multigrid methods have been proposed for SIMD computers [25, 26, 33]. Some of the algorithms are instrinsically parallel [25, 26] or have increased parallelism because they use multiple coarse grids, for example [33]. These efforts and others have been recently reviewed [14. 53, 92]. Most of the new ideas have not been developed yet for solving the incompressible Navier-Stokes equations. One of the most prominent concerns addressed in the literature regarding parallel implementations of serial multigrid methods is the coarse grids. When the number of grid points is smaller than the number of processors the parallelism is reduced to the number of grid points. This loss of parallelism may significantly affect the 17 parallel efficiency. One of the routes around the problem is to use multiple coarse grids [59, 33, 79]. Another is to alter the grid-scheduling to avoid coarse grids. This approach can lead to computationally scalable implementations [34, 49] but may sacrifice the convergence rate. â€œAgglomerationâ€ is an efficiency-increasing technique used in MIMD multigrid programs which refers to the technique of duplicating the coarse grid problem in each processor so that computation proceeds independently (and redundantly). Such an approach can also be scalable [51]. However, most attenÂ¬ tion so far has focused on parallel implementations of serial multigrid algorithms, in particular on assessing the importance of the coarse-grid smoothing problem for difÂ¬ ferent machines and on developing techniques to minimize the impact on the parallel efficiency. 1.6 Description of the Research The dissertation is organized as follows. Chapter 2 discusses the role of the mass conservation in the numerical consistency of the single-grid SIMPLE method for open boundary problems, and explains the relevance of this issue to the convergence rate. In Chapter 3 the single-grid pressure-correction method is implemented on the MP-1, CM-2, and CM-5 computers and its performance is analyzed. High parallel efficienÂ¬ cies are obtained at speeds and problem sizes well beyond the current performance of such algorithms on traditional vector supercomputers. Chapter 4 develops a multigrid numerical method for the purpose of accelerating the single-grid pressure-correction method and maintaining the accelerated convergence property independent of the problem size. The multigrid smoother, the intergrid transfer operators, and the staÂ¬ bilization strategy for Navier-Stokes computations are discussed. Chapter 5 describes the actual implementation of the multigrid algorithm on the CM-5, its convergence rate, and its parallel run time and scalability. The convergence rate depends on the 18 flow problem and the coarse-grid discretization, among other factors. These factors are considered in the context of the â€œfull-multigridâ€ (FMG) starting procedure by which the initial guess on the fine grid is obtained. The cost of the FMG proceÂ¬ dure is a concern for parallel computation [88], and this issue is also addressed. The results indicate that the FMG procedure may influence the asymptotic convergence rate and the stability of the multigrid iterations. Concluding remarks in each chapter summarize the progress made and suggest avenues for further study. 19 Figure 1.1. Staggered-grid layout of dependent variables, for a small but complete domain. Boundary values involved in the computation are shown. Representative u, v, and pressure boundary control volumes are shaded. Front End (CM-2 and MP-1) Partition Manager (CM-5) â€”> serial code, control code, scalar data short blocks of parallel code Sequencer (CM-2) Array control unit (MP-1) Multiple SPARC nodes (CM-5^ more P.E.s â€¢ â€¢ â€¢ array data partitioned among processor memories â€¢ â€¢ â€¢ Interprocessor communication network hypercube (CM-2) + â€œNEWSâ€ 3-stage crossbar (MP-1) + â€œX-Netâ€ fat tree (CM-5) Figure 1.2. Layout of the MP-1, CM-2, and CM-5 SIMD computers. CHAPTER 2 PRESSURE-CORRECTION METHODS 2.1 Finite-Volume Discretization on Staggered Grids The formulation of the numerical method used in this work begins with the inteÂ¬ gration of the governing equations Eq 1.1-1.3 over each of the control volumes in the computational domain. Figure 1.1 shows a model computational domain with u, v, and p (cell-centered) control volumes shaded. The continuity equation is integrated over the p control volumes. Consider the discretization of the u-momentum equation for the control volume shown in Figure 2.1 whose dimensions are Ax and Ay. The v control volumes are done exactly the same except rotated 90Â°. Integration of Eq. 1.2 over the shaded region is interpreted as follows for each of the terms: // apUdxdy='^AxAy, dt dpu2 dt JI dx dy = ~ PU9 Ayâ€™ // dx dpuv dy dx dy = (punvn - pusvs) Ax, // dp dx dxdy = (pe ~ pw)Ay, JJ J! d d2 u dx2 d2u dxJy=U- dn dx du gyldxdy=U dy -pTx f* du dx dii dy Ay Ax (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) The lowercase subscripts e, w. n, s indicate evaluation on the control volume faces. By convention and the mean-value theorem, these are at the midpoint of the faces. The subscript P in Eq. 2.1 indicates evaluation at the center of the control volume. 22 Because of the staggered grid, the required pressure values in Eq. 2.4 are already located on the u control volume faces. The pressure-gradient term is effectively a second-order central-difference approximation. With colocated grids, however, the control-volume face pressures are obtained by averaging the nearby pressures. This averaging results in the pressure at the cell center dropping out of the expression for the pressure gradient. The central-difference in Eq. 2.4 is effectively taken over a distance 2Ax on colocated grids. Thus staggered cartesian grids provide a more accurate approximation of the pressure-gradient term since the difference stencil is smaller. The next step is to approximate the terms which involve values at the control volume faces. In Eq. 2.2, one of the ue and one of the uw are replaced by an average of neighboring values, ( 2 2 [pue - puu \ * / ue + up up + uw i A ) Ay = [ p ue - P ^ Uw ) Ay (2.7) and in Eq. 2.3, vn and vs are obtained by averaging nearby values, / \ a / ^Tie + Vnw vse + Vsw \ A (punvn - pusvs) Ax = ( p un - p r U* ) Ax (2.8) The remaining face velocities in the convection terms, uâ€ž, us, ue. and uw, are exÂ¬ pressed as a certain combination of the nearby u valuesâ€”which u values are involved and what weighting they receive is prescribed by the convection scheme. Some popÂ¬ ular recirculating flow convection schemes are described in [73, 75]. The control-volume face derivatives in the diffusion terms are evaluated by central du du \ Ay = ( ue â€” up ^ Ax e w / du du Ax - ( un up !* dv ^ 9y n Â° J V Ay - p- Ax J ) Ay P- Ay Ax (2.9) (2.10) differences, 23 The unsteady term in Eq. 2.1 is approximated by a backward Euler scheme. All the terms are evaluated at the â€œnewâ€ time level, i.e. implicitly. Thus, the discretized momentum equations for each control volume can be put into the following general form, apup = apup + a\vu\v + aNuN + Â«s^s + (2-11) where b = (pw â€” pe)Ay+pup/ At, the superscript n indicating the previous time-step. The coefficients ap, as, etc. are comprised of the terms which modify u^-, up, etc. in the discretized convection and diffusion terms. The continuity equation is integrated over a pressure control volume, dpu dpv dx + dy Again the staggered grid is an advantage because the normal velocity components on each control volume face are already in positionâ€”there is no need for interpolation. 2.2 The SIMPLE Method One SIMPLE iteration takes initial velocity and pressure fields (u*,v*,p*) and computes new guesses (u,v,p). The intermediate values are denoted with a tilde, (Ã¼,v,p). In the algorithm below, a^(u*,v*), for example, means that the apÂ¡ coeffiÂ¬ cient in the ii-momentum equation depends on u* and v*. The parameters uu, vv, and vc are the numbers of â€œinnerâ€ iterations to be taken for the u. u, and continuity equaÂ¬ tions, respectively. This notation will be clarified by the following discussion. The inner iteration count is indicated by the superscript enclosed in parentheses. Finally, uuv and LOc are the relaxation factors for the momentum and continuity equations. SIMPLE (u*,v*,p*;i/u,i/v,i/p,uuv,uc) Compute u coefficients aj*(u*,u*) (k = P,E,W,N,S) and source term 6u(n*,p*) dx dy = p(ue - uw)Ay + p{vn - us)Aa: = 0. (2.12) 24 for each discrete w-momentum equation: flu au + as^s + o,EÃ¼E + a^uw + bu + (1 - uuv)^uP Do vu iterations to obtain an approximate solution for Ã¼ starting with u* as the initial guess = Gu(n~l'> + fu u = u("=**> Compute v coefficients aÂ£(u,u*) (k = E,W,N,S) and source term 6v(u*,p*) for each discrete r-momentum equation: ^vp = avNvat + avsvs + aEvE + avwvw + bv + (1 - uuv)^v*P Do uv iterations to obtain an approximate solution for v starting with v* as the initial guess v(n) _ Qv(n-1) + fv V = V Compute p1 coefficients ack (k = P,E,W,N,S) and source term 6c(Ã±,Ã±) for each discrete p1 equation: o-pp'p = anP'n + asP's + aeP'e + awP'w + bÂ° Do uc iterations to obtain an approximate solution for p' starting with zero as the initial guess p'(n) _ Qp'(n-1) + fc Correct it, u, and p* at every interior grid point Up = Ã¼p + [p'w-v'^v Up - Up (ou)p ~ I (P'. â€” pâ€™rPAl VP = VP+ (Wp~ Pp = Pp + UcP'p 25 The algorithm is not as complicated as it looks. The important point to note is that the major tasks to be done are the computing of coefficients and the solving of the systems of equations. The symbol G indicates the iteration matrix of whatever type relaxation is used on these inner iterations (SLUR in this case), and / is the corresponding source term. In the SIMPLE pressure-correction method [61], the averages in Eq. 2.7 and 2.8 are lagged in order to linearize the resulting algebraic equations. The governing equations are solved sequentially. First, the u momentum equation coefficients are computed and an updated u field is computed by solving the system of linear algeÂ¬ braic equations. The pressures in Eq. 2.4 are lagged. The v momentum equation is solved next to update v. The continuity equation, recast in terms of pressure correcÂ¬ tions, is then set up and solved. These pressure corrections are coupled to velocity corrections. Together they are designed to correct the velocity field so that it satisfies the continuity constraint, while simultaneously correcting the pressure field so that momentum conservation is maintained. The relationship between the velocity and pressure corrections is derived from the momentum equation, as described in the next section. The resulting system of equations is fully coupled, as one might expect knowing the elliptic nature of pressure in incompressible fluids, and is therefore expensive to solve. However, if the resulting system of pressure-correction equations were solved exactly, the divergence- free constraint and the momentum equations (with old values of u and v present in the nonlinear convection terms) would be satisfied. This approach would constitute an implicit method of time integration for the linearized equations. The time-step size would have to be limited to avoid stability problems caused by the linearization. To reduce the computational cost, the SIMPLE prescription is to use an approxÂ¬ imate relationship between the velocity and pressure corrections (hence the label 26 â€œsemi-implicitâ€). Variations on the original SIMPLE approximation have shown betÂ¬ ter convergence rates for simple flow problems, but in discretizations on curvilinear grids and other problems with significant contributions from source terms, the perÂ¬ formance is no better than the original SIMPLE method (see the results in [4]). The goal of satisfying the divergence-free constraint can still be attained, if the system of pressure-correction equations is converged to strict tolerances, because the discrete continuity equations are still being solved. But satisfaction of the momentum equations cannot be maintained with the approximate relationship. Consequently it is no longer desirable to solve the p'-system of equations to strict tolerances. ItÂ¬ erations are necessary to find the right velocities and pressures which satisfy all three equations. Furthermore, since the equation coefficients are changing from one iteration to the next, it is pointless to solve the momentum equations to strict tolÂ¬ erances. In practice, only a few iterations of a standard scheme such as successive line-underrelaxation (SLUR) are performed. The single â€œouterâ€ iteration outlined above is repeated many times, with underÂ¬ relaxation to prevent the iterations from diverging. In this sense a two-level iterative procedure is being employed. In the outer iterations, the momentum and pressure- correction equations are iteratively updated based on the linearized coefficients and sources, and inner iterations are applied to partially solve the systems of linear algeÂ¬ braic equations. The fact that only a few inner iterations are taken on each system of equations sugÂ¬ gests that the asymptotic convergence rate of the iterative solver, which is the usual means of comparison between solvers, does not necessarily dictate the convergence rate of the outer iterative process. Braaten and Shyy [4] have found that the conÂ¬ vergence rate of the outer iterations actually decreases when the pressure-correction equation is solved to a much stricter tolerance than the momentum equations. They 27 concluded that the balance between the equations is important. Because u, v, and p' are segregated, the overall convergence rate is strongly dependent on the particÂ¬ ular flow problem, the grid distribution and quality, and the choice of relaxation parameters. In contrast to projection methods, which are two-step but treat the convection terms explicitly (or more recently by solving a Riemann problem [2]) and are therefore restricted from taking too large a time-step, the pressure-correction approach is fully implicit with no time-step limitation, but many iterations may be necessary. The projection methods are formalized as time-integration techniques for semi-discrete equations. SIMPLE is an iterative method for solving the discretized Navier-Stokes system of coupled nonlinear algebraic equations. But the details given above should make it clear that these techniques bear strong similaritiesâ€”specifically, a single SIMPLE iteration would be a projection method if the system of pressure-correction equations was solved to strict tolerances at each iteration. It would be interesting to do some numerical comparisons between projection methods and pressure-correction methods to further clarify the similarity. 2.3 Discrete Formulation of the Pressure-Correction Equation The discrete pressure-correction equation is obtained from the discrete momentum and continuity equations as follows. The velocity field which has been newly obtained by solving the momentum equations was denoted by (Ã¼, v) earlier. The pressure field after the momentum equations are solved still has the initial value p*. So ft, i, and p* satisfy the tt-momentum equation apÃ¼p = apiiE + awÃ¼w + cinÃœm + asÃ¼s + [p*w â€” p*)Ay, (2.13) and the corresponding u-momentum equation. The corrected (continuity-satisfying) velocity field (u,v) satisfies the ^-momentum equation with the corrected pressure 28 field p, apup = apup + awuw + aNuN + as^-s + {pw ~ Pe)Ay, (2.14) and likewise for the v-momentum equation. Additive corrections are assumed, i.e. u = Ã¼ + u' (2.15) v = v + V (2.16) p â€” p' + p'. (2.17) Subtracting Eq. 2.13 from Eq. 2.14 gives the desired relationship between pressure and the u corrections, aPu'p = H akUk + (p'w - p'e)Ay, (2.18) k=E,W1N,S with a similar expression for the v corrections. If Eq. 2.18 is used as is, then the nearby velocity corrections in the summation need to be replaced bv similar expressions involving pressure-corrections. This requirement brings in more velocity corrections and more pressure corrections, and so on, leading to an equation which involves the pressure corrections at every grid point. The resulting system of equations would be expensive to solve. Thus, the summation term is dropped in order to obtain a compact expression for the velocity correction in terms of pressure corrections. At convergence, the pressure corrections (and therefore the velocity corrections) go to zero, so the precise form of the approximate pressure- velocity correction relationship does not figure in the final converged solution. The discrete form of the pressure-correction equation follows by first substituting the simplified version of Eq. 2.18 into Eq. 2.15, up = up + u'p = up + (p'w - p')Ay, (2.19) 29 and then substituting this into the continuity equation Eq. 2.12, (with an analogous formula for vp). The result is pAÂ¡J (Pp-p'e)~ P^V Ã‘ {p'w Pp) + pAX .. (Pp â€” Pyv) â€” PAX^PS-P'p) = b (2-20) aP(ue)Krr rtj> aP(uw) where the source term b is aP(vn) ap{vs) b = puwAy â€” pueAy + pv* Ax â€” pv* Ax (2.21) Recall that Eq. 2.20 and Eq. 2.21 are written for the pressure control volumes, so that there is some interpretation required. The term ap(ue) in Eq. 2.20 is the appropriate ap for the discretized u-momentum equation, Eq. 2.13. In other words, up in Eq. 2.13 is actually ue, uw, un, or us in Eq. 2.20 and 2.21, relative to the pressure control volumes on the staggered grid. Eq. 2.20 can be rearranged into the same general form as Eq. 2.11. From Eq. 2.21, it is apparent that the right-hand side term is the net mass flux entering the control volume, which should be zero in incompressible flow. In the formulation of the pressure-correction equation for boundary control volÂ¬ umes, one makes use of the fact that the normal velocity components on the boundÂ¬ aries are known from either Dirichlet or Neumann boundary conditions, so no velocity correction is required there. Consequently, the formulation of Eq. 2.20 for boundary control volumes does not require any prescription of boundary p' values [60] when velocity boundary conditions are prescribed. Without the summation from Eq. 2.18, it is apparent that a zero velocity correction for the outflow boundary u-velocity component is obtained when pw = peâ€”in effect, a Neumann boundary condition on pressure is implied. This boundary condition is appropriate for an incompressible fluid because it is physically consistent with the governing equations in which only the pressure gradient appears. There is a unique pressure gradient but the level is 30 adjustable by any constant amount. If it happens that there is a pressure specified on the boundary, for example by Eq. 1.4, then the correction there will be zero, proÂ¬ viding a boundary condition for Eq. 2.20. Thus, it seems that there are no concerns over the specification of boundary conditions for the p' equations. 2.4 Well-Posedness of the Pressure-Correction Equation 2.4.1 Analysis To better understand the characteristics of the pressure-correction step in the SIMPLE procedure, consider a model 3x3 computational domain, so that 9 algebraic equations for the pressure corrections are obtained. Number the control volumes as shown in Figure 2.3. Then the system of p' equations can be written CL p ~aE 0 ~aN 0 0 0 0 0 P'l p(ul - + v}, -Â«Â¡n ~aw a2P ~aE 0 ~aN 0 0 0 0 P'l P(Ue n i 2 - uw + vn -vl) 0 ~aw flp 0 0 ~aN 0 0 0 p'z P(Â«e â€” U^ -4- uw ' un -Â«?) ~as 0 0 dp aE 0 ~aN 0 0 Pa P(u4e ~ + V4 -V4) 0 ~as 0 ~aw (Vp ~aE 0 ~aN 0 Ã = P(ul ~ Â«5, + Â»n -v5,) 0 0 â€” as 0 ~aw Clp 0 0 ~aN Pe P(Ue - + Â«6 -v6s) 0 0 0 -4 0 0 CL p 7 -aâ€˜E 0 Pi p(K -ul + v7n -V7,) 0 0 0 0 -<4 0 â€” aw asp -a| P's P{< ~ < + < -Vs,) 0 0 0 0 0 -a| 0 Q ~aw Op .P9. Lp(Â«e - u9w + -oj where the superscript designates the cell location and the subscript designates the coefficient linking the point in question, P, and the neighboring node. The right-hand side velocities are understood to be tilde quantities as in Eq. 2.21. In finite-volume discretizations, fluxes are estimated at the control volume faces which are common to adjacent control volumes, so if the governing equations are cast in conservation law form, as they are here, the discrete efflux of any quantity out of one control volume is guaranteed to be identical to the influx into its neighbor. There is no possibility of internal sources or sinks. In fact this is what makes finite- volume discretizations preferable to finite-difference discretizations. The following 31 relationships, using control volume 5 in Figure 2.3 as an example, follow from Eq. 2.20 and the internal consistency of finite-volume discretizations: Op â€” Op -(- aw + a| a)y, (2.23) aW aEi aE â€” aWi 4 = 4i 4 â€” aN (2.24) 4 = ul 4 = u6w, vl = vl v5s = vl (2.25) Eq. 2.23 states that the coefficient matrix is pentadiagonal and diagonally dominant for the interior control volumes. Furthermore, when the natural boundary condition (zero velocity correction) is applied, the appropriate term in Eq. 2.20 for the boundary under consideration does not appear, and therefore the pressure-correction equations for the boundary control volumes also satisfy Eq. 2.23. If a pressure boundary condiÂ¬ tion is applied so that the corresponding pressure correction is zero, then one would set pp = 0 in Eq. 2.20, for example, which would give aw + aN + as < ap. Thus, either way, the entire coefficient matrix in Eq. 2.22 is diagonally dominant. However, with the natural prescription for boundary treatment, no diagonal term exceeds the sum of its off-diagonal terms. Thus, the system of equations Eq. 2.22 is linearly dependent with the natural (velocity) boundary conditions, which can be verified by adding the 9 equations above. Because of Eq. 2.23 and Eq. 2.24 all terms on the left-hand side of Eq. 2.22 identically cancel one another. At all interior control volume interfaces, the right- hand side terms identically cancel due to Eq. 2.25, and the remaining source terms are simply the boundary mass fluxes. This cancellation is equivalent to a discrete statement of the divergence theorem f V - u dtt = / Jn Jan u â– n d{dÃœ) (2.26) 32 where ft is the domain under consideration and n is the unit vector in the direction normal to its boundary dfl. Due to the linear dependence of the left-hand side of Eq. 2.22, the boundary mass fluxes must also sum to zero in order for the system of equations to be consistent. No solution exists if the linearly dependent system of equations is inconsistent. The situation can be likened to a steady-state heat conduction problem with source terms and adiabatic boundaries. Clearly, a steady-state solution only exists if the sum of the source terms is zero. If there is a net heat source, then the temperature inside the domain will simply rise without bound if an iterative solution strategy (quasi time-marching) is used. Likewise, the net mass source in flow problems with open boundaries must sum to zero for the pressure-correction equation to have a solution. In other words, global mass conservation is required in discrete form in order for a solution to exist. The interesting point to note is that during the course of SIMPLE iterations, when the pressure-correction equation is executed, the velocity field does not usually conserve mass globally in flow problems with open boundaries, unless explicit measure is taken to enforce global mass conservation. The purpose of solving the pressure-correction equations is to drive the local mass sources to zero by suitable velocity corrections. But the pressure-correction equations which are supposed to accomplish this purpose do not have a solution unless the net mass source is already zero. For domains with closed boundaries, global mass conservation is obviously not an issue. Furthermore, this problem does not only show up when the initial guess is bad. In the backward-facing step flow discussed below, the initial guess is zero everywhere except for inflow, which obviously is the worst case as far as a net mass source is concerned (all inflow and no outflow). But even if one starts with a mass-conserving initial guess, during the course of iterations the outflow velocity boundary condition 33 which is necessary to solve the momentum equations will reset the outflow so that the global mass-conservation constraint is violated. 2.4.2 Verification bv Numerical Experiments Support for the preceding discussion is provided by numerical simulation of two model problems, a lid-driven cavity flow and a backward-facing step flow. The conÂ¬ figurations are shown along with other relevant data in Figure 2.2. Figure 2.4 shows the outer-loop convergence paths for the lid-driven cavity flow and the backward-facing step flow, both at Re = 100. The quantities plotted in Figure 2.4 are the logw of the global residuals for each governing equation obtained by summing up the local residuals, each of which is obtained by subtracting the left-hand side of the discretized equations from the right-hand side. For the cavity flow there are no mass fluxes across the boundary so, as mentioned earlier, the global mass conservation condition is always satisfied when the algorithm reaches the point of solving the system of //-equations. The residuals have dropped to 10â€' after 150 iterations, which is very rapid convergence, indicating that good pressure and velocity corrections are being obtained. In the backward-facing step flow, however, the flowfield is very slow to develop because no global mass conservation measure is enforced. During the course of iterÂ¬ ations, the mass flux into the domain from the left is not matched by an equal flux through the outflow boundary, and consequently the system of pressure-correction equations which is supposed to produce a continuity-satisfying velocity field does not have a solution. Correspondingly one observes that the outer-loop convergence rate is about 10 times worse than for cavity flow. Also, note that the momentum convergence path of the backward-facing step flow in Figure 2.4 tends to follow the continuity equation, indicating that the pressure and 34 velocity fields are strongly coupled. The present flow problem bears some similarity to a fully-developed channel flow, in which the streamwise pressure-gradient and crossÂ¬ stream viscous diffusion are balanced, so the observation that pressure and velocity are strongly coupled is intuitively correct. Thus, the convergence path is controlled by the development of the pressure field. The slow convergence rate problem is due to the inconsistency of the system of pressure-correction equations. The inner-loop convergence path (the SLUR iterations) for the p'-system of equaÂ¬ tions must be examined to determine the manner in which the inner-loop inconsisÂ¬ tency leads to poor outer-loop convergence rates. Table 2.1 shows leading eigenvalues for successive line-underrelaxation iteration matrices of the p'-system of equations at an intermediate iteration for which the outer-loop residuals had dropped to approxÂ¬ imately 10-2. Largest 3 eigenvalues Cavity Flow Back-Step Flow Ai 1.0 1.0 a2 0.956 0.996 ^3 0.951 0.984 Table 2.1. Largest eigenvalues of iteration matrices during an intermediate iteraÂ¬ tion, applying the successive line-underrelaxation iteration scheme to the p'-system of equations. In both model problems the spectral radius is 1.0 because the p'-system of equaÂ¬ tions is linearly dependent. The next largest eigenvalue is smaller in the cavity flow computation than in the step flow computation, which means a faster asymptotic conÂ¬ vergence rate. However, the difference between 0.996 and 0.956 is not large enough to produce the significant difference observed in the outer convergence path. Figure 2.5 shows the inner-loop residuals of the SLUR procedure during an interÂ¬ mediate iteration. The two momentum equations are well-conditioned and converge to a solution within 4 iterations. In Figure 2.5 for the cavity flow case, the p'-equation 35 converges to zero, although this happens at a slower rate than the two momentum equations because of the diffusive nature of the equation. In Figure 2.5 for the back- step flow, the inner-loop residual is fixed on a nonzero residual, which is in fact the initial level of inconsistency in the system of equations, i.e. the global mass deficit. Given that the system of p'- equations which is being solved does not satisfy the global continuity constraint, however, the significance or utility of the p'-field that has been obtained is unknown. In practice, the overall procedure may still be able to lead to a converged soluÂ¬ tion, as in the present case. It appears that the outflow extrapolating procedure, a zero-gradient treatment utilized here, can help induce the overall computation to converge to the right solution [72]. Obviously, such a lack of satisfaction of global mass conservation is not desirable in view of the slow convergence rate. Further study suggests that the iterative solution to the inconsistent system of p'-equations converges on a unique pressure gradient, i.e. the difference between p' values at any two points tends to a constant value, even though the p'-field does not in general satisfy any of the equations in the system. This relationship is shown in Figure 2.6, in which the convergence of the difference in p' between the lower-left and upper-right locations in the domain of the cavity and backward-facing step flows is plotted. Also shown is the value of p' at the lower-left corner of the domain. For the cavity flow, there is a solution to the system of p'-equations, and it is obtained by the SLUR technique in about 10 iterations. Thus all the pressure corrections and the differences between them tend towards constant values. In the backward-facing step flow, however, the individual pressure corrections increase linearly with the number of iterations, symptomatic of the inconsistency in the system of equations. The differences between p' values approach a constant, however. The rate at which this 36 unique pressure-gradient field is obtained depends on the eigenvalues of the iteration matrix. To resolve the inconsistency problem in the p'-system of equations and thereby improve the outer-loop convergence rate in the backward-facing step flow, global mass conservation has been explicitly enforced during the sequential solution procedure. The procedure used is to compute the global mass deficit and then add a constant value to the outflow boundary u-velocities to restore global mass conservation. AlÂ¬ ternatively, corrections can be applied at every streamwise location by considering control volumes whose boundaries are the inflow plane, the top and bottom walls of the channel, and the i=constant line at the specified streamwise location. The artificially-imposed convection has the effect of speeding up the development of the pressure field, whose normal development is diffusion-dominated. It is interesting to note that this physically-motivated approach is in essence an acceleration of converÂ¬ gence of the line-iterative method via the technique called additive correction [45, 69]. The strategy is to adjust the residual on the current line to zero by adding a conÂ¬ stant to all the unknowns in the line. This procedure is done for every line, for every iteration, and generally produces improvement in the SLUR solution of a system of equations. Kelkar and Patankar [45] have gone one step further by applying additive corrections like an injection step of a multigrid scheme, a so-called block correction technique. This technique is exploited to its fullest by Hutchinson and Raithby [42]. Given a fine-grid solution and a coarse grid, discretized equations for the correction quantities on the coarse grid are obtained by summing the equations for each of the fine-grid cells within a given coarse grid cell. A solution is then obtained (by direct methods in [45]) which satisfies conservation of mass and momentum. The corrections are then distributed uniformly to the fine grid cells which make up the coarse grid 37 cell, and the iterative solution on the fine grid is resumed. However, experiences have shown that the net effect of such a treatment for complex flow problems is limited. Figure 2.7 illustrates the improved convergence rate of the continuity equation for the inner and outer loops, in the backward-facing step flow, when conservation of mass is explicitly enforced. The inner-loop data is from the 10th outer-loop iteration. In Figure 2.7, the cavity flow convergence path is also shown to facilitate the comparison. For the back-step, the overall convergence rate is improved by an order of magnitude, becoming slightly faster than the cavity flow case. This result reflects the improved inner-loop performance, also shown in Figure 2.7. The improved performance for the pressure-correction equation comes at the expense of a slightly slower convergence rate for the momentum equations, because of the nonlinear convection term. In short, it has been shown that a consistency condition, which is physically the reÂ¬ quirement of global mass conservation, is critical for meaningful pressure-corrections to be guaranteed. Given natural (velocity) boundary conditions, which lead to a linearly dependent system of pressure-correction equations, satisfaction of the global continuity constraint is the only way that a solution can exist, and therefore the only way that the inner-loop residuals can be driven to zero. For the model backwardÂ¬ facing step flow in a channel with length L = 4 and a 21 x 9 mesh, the mass- conservation constraint is enforced globally or at every streamwdse location by an additive-correction technique. This technique produces a 10-fold increase in the conÂ¬ vergence rate. Physically, modifying the u velocities has the same effect as adding a convection term to the Poisson equation for the //-field, which otherwise develops very slowly. A coarse grid size was used to demonstrate the need of enforcing global mass conservation. On a finer grid, this issue becomes more critical. In the next section, the solution accuracy aspects related to mass conservation will be addressed, and the computations will be conducted with more adequate grid resolution. 38 2.5 Numerical Treatment of Outflow Boundaries Continuing with the theme of well-posedness, the next numerical issue to be disÂ¬ cussed is the choice of outflow boundary location. If fluid flows into the domain at a boundary where extrapolation is applied, then, traditionally, the problem is not considered to be well-posed, because the information which is being transported into the domain does not participate in the solution to the problem [60]. Numerically, however, accurate solutions can be obtained using first-order extrapolation for the veÂ¬ locity components on a boundary where inflow is occurring [72]. Here open boundary treatment for both steady and time-dependent flow problems is investigated further. Figure 2.9 and 2.8 present streamfunction contours for a time-dependent flow problem, impulsively started backward-facing step flow, using central-differencing for the convection terms and first-order backward-differencing in time. A parabolic inflow velocity profile is specified, while outflow boundary velocities are obtained by first-order extrapolation. The Reynolds number based on the average inflow velocity uavg and the channel height H, is 800. The expansion ratio H/h is 2 as in the model problem described in Figure 2.3. Time-accurate simulations were performed for two channel configurations, one with length L = 8 (81 x 41 mesh) and the other with length L = 16 (161 x 41 mesh). This flow problem has been the subject of some recent investigations focusing on open boundary conditions [30, 31]. For each time step, the SIMPLE algorithm is used to iteratively converge on a solution to the unsteady form of the governing equations, explicitly enforcing global conservation of mass during the course of iterations. In the present study, convergence was declared for a given time step when the global residuals had been reduced below 10-4. The time-step size was twice the viscous time scale in the y-direction, i.e. 39 Ai = 2Ay2/u. Thus a fluid particle entering the domain at the average velocity u = 1 travels 2 units downstream during a time-step. Figure 2.8 shows the formation of alternate bottom/top wall recirculation regions during startup which gradually become thinner and elongated as they drift downÂ¬ stream. For the L = 16 simulation (Figure 2.8), the transient flowfield has as many as four separation bubbles at T = 32, the latter two of which are eventually washed out of the domain. In the L = 8 simulation (Figure 2.9) the streamfunction plots are at times corresponding to those shown in Figure 2.8. Note that between T = 11 and T = 32, a secondary bottom wall recirculation zone forms and drifts downstream, exiting without reflection through the downstream boundary. The time evolution of the flowfield for the L = 8 and L â€” 16 simulations is virtually identical. As can be observed, the facts that a shorter channel length was used in Figure 2.9 and that a recirculating cell may go through the open boundary do not affect the solutions. Figure 2.10 compares the computed time histories of the bottom wall reattachment and top wall separation points between the two computations. The L â€” 8 and L = 16 curves are perfectly overlapped. The steady-state solutions for both the L = 8 and L = 16 channel configurations are also shown in Figure 2.9 and 2.8, respectively. Although the outflow boundary cuts the top wall separation bubble approximately in half, there is no apparent difference between the computed streamfunction contours for 0 < x < 8. Furthermore, the convergence rate is not affected by the choice of outflow boundary location. Figure 2.11 compares the steady-state u and v velocity profiles at x = 7 beÂ¬ tween the two computations. The accuracy of the computed results is assessed by comparison with an FEM numerical solution reported by Gartling [27]. Figure 2.11 establishes quantitatively that the two simulations differ negligibly over 0 < x < 8 (the v profile differs on the order of 10-3) The velocity scale for the problem is 1. 40 Neither v profile agrees perfectly with the solution obtained by Gartling, which may be attributed to the need for conducting further grid refinement studies in the present work and/or Gartlingâ€™s work. Evidently the location of the open boundary is not critical to obtaining a conÂ¬ verged solution. This observation indicates that the downstream information is comÂ¬ pletely accounted for by the continuity equation. The correct pressure field can deÂ¬ velop because the system of //-equations requires only the boundary mass flux specifiÂ¬ cation. If the global continuity constraint is satisfied, the pressure-correction equation is consistent regardless of whether there is inflow or outflow at the boundary where extrapolation is applied. The numerical well-posedness of the open boundary comÂ¬ putation results in virtually identical flowfield development for the time-dependent L = 8 and L = 16 simulations as well as steady-state solutions which agree with each other and follow closely Gartlingâ€™s benchmark data [27]. 2.6 Concluding Remarks In order for the SIMPLE pressure-correction method to be a well-posed numerÂ¬ ical procedure for open boundary problems, explicit steps must be taken to ensure the numerical consistency of the pressure-correction system of equations during the course of iterations. For the discrete problem with the natural boundary treatment for pressure, i.e. normal velocity specified at all boundaries, global mass conservaÂ¬ tion is the solvability constraint which must be satisfied in order that the system of p'-equations is consistent. Without a globally mass-conserving procedure enforced during each iterative step, the utility of the pressure-corrections obtained at each itÂ¬ eration cannot be guaranteed. Overall convergence may still occur, albeit very slowly. In this regard, the poor outer-loop convergence behavior simply reflects the (poor) convergence rate of the inner-loop iterations of the SLUR technique. In general, the 41 inner-loop residual is fixed on the value of the initial level of inconsistency of the system of p'-equations which physically is the global mass deficit. The convergence rate can be improved dramatically by explicitly enforcing mass conservation using an additive-correction technique. The results of numerical simulations of backward- facing step flow illustrate and support these conclusions. The mass-conservation constraint also has implications for the issue of proper numerical treatment of open boundaries where inflow is occurring. Specifically, the conventional viewpoint that inflow cannot occur at open boundaries without Dirich- let prescription of the inflow variables can be rebutted, based on the grounds that the numerical problem is well-posed if the normal velocity components satisfy the continuity constraint. 42 Figure 2.1. Staggered grid u control volume and the nearby variables which are involved in the discretization of the u-momentum equation. 43 U= 1 U(y) St. St Nk N \v > Vi \ s. s*. X V V \ \ \ St, s, 5 1 S. V4 \ A V > \ 's*, V \ A \ V \ Nk ~s*. â– N. \ \ \ \ H Figure 2.2. Description of two model problems. Both are at Re â€” 100. The cavity is a square with a top wall sliding to the left, while the backward-facing step is a 4x1 rectangular domain with an expansion ratio H/h â€” 2, and a parabolic inflow (average inflow velocity =1). The cavity flow grid is 9 x 9 and the step flow grid is 21 x 9. The meshes and the velocity vectors are shown. 44 Figure 2.3. Model 3x3 computational domain with numbered control volumes, for discussion of Eq. 2.22. The staggered velocity components which refer to control volume 5 are also indicated. Log10 of Residual 45 Re = 100 Cavity Flow Re = 100 Back-Step Flow # of Iterations # of Iterations Figure 2.4. Outer-loop convergence paths for the Re = 100 lid-driven cavity and backward-facing step flows, using central-differencing for the convection terms. LegÂ¬ end: p' equation: u momentum equation: v momentum equation. 46 # of Iterations # of Iterations Figure 2.5. Inner-loop convergence paths for the Re = 100 lid-driven cavity and backward-facing step flows. The vertical axis is the log\o of the ratio of the current residual to the initial residual. Legend: p' equation: u momentum equation: v momentum equation. 47 Inner Loop for Cavity Flow Inner Loop for Back-Step Flow Figure 2.6. Variation of p' with inner-loop iterations. The dashed line is the value of p at the lower-left control volume, while the solid line is the difference between Plowerleft an<^ Pupperrighf 48 Outer Loop Convergence Path Inner-Loop Convergence Path # of Iterations # of Iterations Figure 2.7. Outer-loop and inner-loop convergence paths of the p' equation for the backward-facing step model problem, with and without enforcing the continuity conÂ¬ straint. (1) conservation of mass not enforced: (2) continuity enforced globally; (3) cavity flow. T= 15 T = 20 T = 32 T z GC Figure 2.8. Time-dependent flowfield for impulsively started backward-facing step flow, Re = 800. I lie domain has length /, = 1G. Streamfnnction contours are plotted at several instants during the evolution to the steady- state, which is the last figure. 50 T = 15 T = 20 T = 32 Tzz 00 Figure 2.9. Time-dependent flowfield for impulsively started backward-facing step flow. Re = 800. The domain has length L = 8. Streamfunction contours are plotted at several instants during the evolution to the steady-state, which is the last figure. 51 Time-Evolution of Reattachment/Separation Locations Time Figure 2.10. Time-dependent location of bottom wall reattachment point and top wall separation point for Re = 800 impulsively started backward-facing step flow. The curves for both L = 8 and L â€” 16 computations are shown; they overlap identically. 52 V Velocity Profile at X = 7 For Re = 800 Back-Step Flow Figure 2.11. Comparison of u and u-component of velocity profiles at x = 7.0 for the L = 16 and L â€” 8 backward-facing step simulations at Re = 800, with central- differencing. (o) indicates the grid-independent FEM solution obtained by Gartling. The v profile is scaled up by 103. CHAPTER 3 EFFICIENCY AND SCALABILITY ON SIMD COMPUTERS The previous chapter considered an issue which was important because of its imÂ¬ plications for the convergence rate in open boundary problems. The present chapter shifts gears to focus on the cost and efficiency of pressure-correction methods on SIMD computers. As discussed in Chapter 1, the eventual goal is to understand the indirect cost [23], i.e. the parallel run time, of such methods on SIMD computers, and how this cost scales with the problem size and the number of processors. The run time is just the number of iterations multiplied by the cost per iteration. This chapter considers the cost per iteration. 3.1 Background The discussion of SIMD computers in Chapter 1 indicated similarities in the general layout of such machines and in the factors which affect program performance. More detail is given in this section to better support the discussion of results. 3.1.1 Speedup and Efficiency Speedup S is defined as s = y< (3.i) ip where Tp is the measured run time using np processors. In the present work Tj is the run time of the parallel algorithm on one processor, including both serial and parallel computational work, but excluding the front-end-to-processor and interproÂ¬ cessor communication. On a MIMD machine it is sometimes possible to actually time 53 54 the program on one processsor, but each SIMD processor is not usually a capable serial computer by itself, so 7\ must be estimated. The timing tools on the CM-2 and CM-5 are very sophisticated, and can separately measure the time elapsed by the processors doing computation, doing various kinds of communication, and doing nothing (waiting for an instruction from the front-end, which might be finishing up some serial work before it can send another code block). Thus, it is possible to make a reasonable estimate for T\. Parallel efficiency is the ratio of the actual speedup to the ideal (np), which reflects the overhead costs of doing the computation in parallel: Sgctual _ T\/Tp Sideal Tip (3.2) If Tcornp is the time in seconds spent by each of the np processors doing useful work (computation), T,nier_proc is the time spent by the processors doing interprocessor communication, and T/e_<0_pr0C is the time elapsed through front-end-to-processor communication, then each of the processors is busy a total of Tcomp + 7Â¿â€žÃer_proc seconds and the total run time on multiple processors is Tcomp + Tinter-proc-\-Tfe-to-proc seconds. Assuming that the parallelism is high, i.e. a high percentage of the virtual processors are not idle, a single processor would need npTcomp time to do the same work. Thus, T\ = npTcomp, and from Eq. 3.2 E can be expressed as 1 1 (3.3) 1 + {Tinier â€” proc + Tfe â€”toâ€”proc) Â¡Tcomp 1 T (TComm ) /Tcomp Since time is work divided by speed, E depends on both machine-related factors and the implementational factors through Eq. 3.3. High parallel efficiency is not necesÂ¬ sarily a product of fast processors or fast communications considered alone, instead it is the xâ€™elative speeds that are important, and the relative amount of communication and computation in the program. Consider the machine-related factors first. 55 3.1.2 Comparison Between CM-2. CM-5, and MP-1 A 32-node CM-5 with vector units, a 16k processor CM-2, and a lk processor MP-1 were used in the present study. The CM-5 has 4 GBytes total memory, while the CM-2 has 512 Mbytes, and the MP-1 has 64 MBytes. The peak speeds of these computers are 4. 3.5, and 0.034 Gflops, respectively, in double precision. Per procesÂ¬ sor, the peak speeds are 32, 7, and 0.033 Mflops, with memory bandwidths of 128, 25, and 0.67 Mbytes/s [67, 83]. Clearly these are computers with very different capaÂ¬ bilities, even taking into account the fact that peak speeds, which are based only on the processor speed under ideal conditions, are not an accurate basis for comparison. In the CM-2 and CM-5 the front-end computers are Sun-4 workstations, while in the MP-1 the front-end is a Decstation 5000. From Eq. 3.3, it is clear that the relative speeds of the front-end computer and the processors are important. Their ratio determines the importance of the front-end-to-processor type of communication. On the CM-2 and MP-1, there is just one of these intermediate processors, called either a sequencer or an array control unit, respectively, while on the 32-node CM-5 the 32 SPARC microprocessors have the role of sequencers. Each SPARC node broadcasts to four vector units (VUs) which actually do the work. Thus a 32-node CM-5 has 128 independent processors. In the CM-2 the â€˜â€˜proÂ¬ cessorsâ€™â€™ are more often called processing elements (PEs), because each one consists of a floating-point unit coupled with 32 bit-serial processors. Each bit-serial processor is the memory manager for a single bit of a 32-bit word. Thus, the 16k-processor CM-2 actually has only 512 independent processing elements. This strange CM-2 processor design came about basically as a workaround which was introduced to imÂ¬ prove the memory bandwidth for floating-point calculations [66]. Compared to the CM-5 VUs, the CM-2 processors are about one-fourth as fast, with larger overhead 56 costs associated with memory access and computation. The MP-1 has 1024 4-bit processorsâ€”compared to either the CM-5 or CM-2 processors, the MP-1 processors are very slow. The generic term â€œprocessing elementâ€ (PE), which is used occassion- ally in the discussion below, refers to either one of the VUs, one of the 512 CM-2 processors, or one of the MP-1 processors, whichever is appropriate. For the present study, the processors are either physically or logically imagined to be arranged as a 2-d mesh, which is a layout that is well-supported by the data networks of each of the computers. The data network of the 32-node CM-5 is a fat tree of height 3, which is similar to a binary tree except the bandwidth stays constant upwards from height 2 at 160 MBytes/s (details in [83]). One can expect approximately 480 MBytes/s for regular grid communication patterns (i.e. between nearest-neighbor SPARC nodes) and 128 MBytes/s for random (global) communicaÂ¬ tions. The randomly-directed messages have to go farther up the tree, so they are slower. The CM-2 network (a hypercube) is completely different from the fat-tree netÂ¬ work and its performance for regular grid communication between nearest-neighbor processors is roughly 350 MBytes/s [67]. The grid network on the CM-2 is called NEWS (North-East-West-South). It is a subset of the hypercube connections seÂ¬ lected at run time. The MP-1 has two networks: regular communications use X-Net (1.25 GBytes/s, peak) which connects each processor to its eight nearest neighbors, and random communications use a 3-stage crossbar (80 MBytes/s, peak). To summarize the relative speeds of these three SIMD computers it is sufficient for the present study to observe that the MP-1 has very fast nearest-neighbor comÂ¬ munication compared to its computational speed, while the exact opposite is true for the CM-2. The ratio of nearest-neighbor communication speed to computation speed is smaller still for the CM-5 than the CM-2. Again, from Eq. 3.3, one expects that these differences will be an important factor influencing the parallel efficiency. 57 3.1.3 Hierarchical and Cut-and-Stack Data Mappings When there are more array elements (grid points) than processors, each processor handles multiple grid points. Which grid points are assigned to which processors is determined by the â€œdata-mapping,â€ also called the data layout. The processors repeat any instructions the appropriate number of times to handle all the array elements which have been assigned to it. A useful idealization for SIMD machines, however, is to pretend there are always as many processors as grid points. Then one speaks of the â€œvirtual processorâ€ ratio (VP) which is the number of array elements assigned to each physical processor. The way the data arrays are partitioned and mapped to the processors is a main concern for developing a parallel implementation. The layout of the data determines the amount of communication in a given program. When the virtual processor ratio is 1, there are an equal number of processors and array elements and the mapping is just one-to-one. When VP > 1 the mapping of data to processors is either â€œhierarchical,â€ in CM-Fortran, or â€œcut-and-stackâ€ in MP-Fortran. These mappings are also termed â€œblockâ€ and â€œcyclicâ€ [85], respectively, in the emerging High-Performance Fortran standard. The relative merits of these different approaches have not been completely explored yet. In cut-and-stack mapping, nearest-neighbor array elements are mapped to nearest- neighbor physical processors. When the number of array elements exceeds the numÂ¬ ber of processors, additional memory layers are created. VP is just the number of memory layers. In the general case, nearest-neighbor virtual processors (i.e. array elements) will not be mapped to the same physical processor. Thus, the cost of a nearest-neighbor communication of distance one will be proportional to VP, since the nearest-neighbors of each virtual processor will be on a different physical processor. In the hierarchical mapping, contiguous pieces of an array (â€œvirtual subgridsâ€) are 58 mapped to each processor. The â€œsubgrid sizeâ€ for the hierarchical mapping is synÂ¬ onymous with VP. The distinction between hierarchical and cut-and-stack mapping is clarified by Figure 3.1. In hierarchical mapping, for V P > 1, each virtual processor has nearest-neighbors in the same virtual subgrid, that is, on the same physical processor. Thus, for hierÂ¬ archical mapping on the CM-2, interprocessor communication breaks down into two types (with different speeds)â€”on-processor and off-processor. Off-processor commuÂ¬ nication on the CM-2 has the NEWS speed given above, while on-processor communiÂ¬ cation is somewhat faster, because it is essentially just a memory operation. A more detailed presentation and modelling of nearest-neighbor communication costs for the hierarchical mapping on the CM-2 is given in [3]. The key idea is that with hierarÂ¬ chical mapping on the CM-2 the relative amount of on-processor and off-processor communication is the area to perimeter ratio of the virtual subgrid. For the CM-5, there are three types of interprocessor communication: (1) between virtual processors on the same processor (that is, the same VU), (2) between virtual processors on different VUs but on the same SPARC node, and (3) between virtual processors on different SPARC nodes. Between different SPARC nodes (number 2), the speed is 480 MBytes/s as mentioned above. On the same VU the speed is 16 GBytes/s. (The latter number is just the aggregate memory bandwidth of the 32- node CM-5.) Thus, although off-processor NEWS communication is slow compared to computation on the CM-2 and CM-5, good efficiencies can still be achieved as a consequence of the data mapping which allows the majority of communication to be of the on-processor type. 59 3.2 Implementional Considerations The cost per SIMPLE iteration depends on the choice of relaxation method (solver) for the systems of equations, the number of inner iterations (z/u, ty, and uc), the computation of coefficients for each system of equations, the correction step, and the convergence checking and serial work done in program control. The pressure- correction equation, since it is not underrelaxed, typically needs to be given more iterations than the momentum equations, and consequently most of the effort is exÂ¬ pended during this step of the SIMPLE method. This is another reason why the convergence rate of the p'-equations discussed in Chapter 2 is important. Typically uu and uv are the same and are < 3, and uc < bvu. In developing a parallel implementation of the SIMPLE algorithm, the first conÂ¬ sideration is the method of solving the u, v, and p' systems of equations. For serial computations, successive line-underrelaxation using the tridiagonal matrix algorithm (TDMA, whose operation count is O(N)) is a good choice because the cost per itÂ¬ eration is optimal and there is long-distance coupling between flow variables (along lines), which is effective in promoting convergence in the outer iterations. The TDMA is intrinsically serial. For parallel computations, a parallel tridiagonal solver must be used (parallel cyclic reduction in the present work). In this case the cost per itÂ¬ eration depends not only on the computational workload (0(Nlog2N)) but also on the amount of communication generated by the implementation on a particular maÂ¬ chine. For these reasons, timing comparisons are made for several implementations of both point- and line-Jacobi solvers used during the inner iterations of the SIMPLE algorithm. 60 Generally, point-Jacobi iteration is not sufficiently effective for complex flow probÂ¬ lems. However, as part of a multigrid strategy, good convergence rates can be obÂ¬ tained (see Chapters 4 and 5). Furthermore, because it only involves the fastest type of interprocessor communication, that which occurs between nearest-neighbor proÂ¬ cessors, point-Jacobi iteration provides an upper bound for parallel efficiency, against which other solvers can be compared. The second consideration is the treatment of boundary computations. In the present implementation, the coefficients and source terms for the boundary control volumes are computed using the interior control volume formula and mask arrays. Oran et al. [57] have called this trick the uniform boundary condition approach. All coefficients can be computed simultaneously. The problem with computing the boundary coefficients separately is that some of the processors are idle, which deÂ¬ creases E. For the CM-5, which is â€œsynchronized MIMDâ€ instead of strictly SIMD, there exists limited capability to handle both boundary and interior coefficients siÂ¬ multaneously without formulating a single all-inclusive expression. However, this capability cannot be utilized if either the boundary or interior formulas involve inÂ¬ terprocessor communication, which is the case here. As an example of the uniform approach, consider the source terms for the north boundary u control volumes, which are computed by the formula b = aNuN + (pw - pe)Ay (3.4) Recall that a# represents the discretized convective and diffusive flux terms, and un is the boundary value, and in the pressure gradient term, Ay is the vertical dimension of the u control volume and pw/pe are the west/east u-control-volume face pressures on the staggered grid. Similar modifications show up in the south, east, and west boundary u control volume source terms. To compute the boundary and interior 61 source terms simultaneously, the following implementation is used: ^ â€” Q'boundaryV'boundary T (Pw Pe)^V (3.5) where uboundary = UN^N + UsLs + Ue^E + Â«wAv (3-6) and dboundary = CLnIn + CLS^S + CLE^E + CL\V I\V (3-7) /jv, Is, Ie-i and Iw are the mask arrays, which have the value 1 for the respective boundary control volumes and 0 everywhere else. They are initialized once, at the beginning of the program. Then, every iteration, there are four extra nearest-neighbor communications. A comparison of the uniform approach with an implementation that treats each boundary separately is discussed in the results. 3.3 Numerical Experiments The SIMPLE algorithm for two-dimensional laminar flow has been timed on a range of problem sizes from 8 x 8 to 1024 x 1024 which, on the CM-5, covers up to VP = 8192. The convection terms are central-differenced. A fixed number (100) of outer iterations are timed using as a model flow problem the lid-driven cavity flow at Re = 1000. The timings were made with the â€œPrismâ€ timing utility on the CM-2 and CM-5, and the â€œdpuTimerâ€ routines on the MP-1 [52, 86]. These utilities can be inaccurate if the front-end machine is heavily loaded, which was the case with the CM-2. Thus, on the CM-2 all cases were timed three times and the fastest times were used, as recommended by Thinking Machines [82]. Prism times every code block and accumulates totals in several categories, including computation time for the nodes (Tcomp), â€œNEWSâ€ communication (Tnews), and irregular-pattern â€œSENDâ€ communication. Also it is possible to infer Tfe-to-proc from the difference 62 between the processor busy time and the elapsed time. In the results Tcomm is the sum of the â€œNEWSâ€ and â€œSENDâ€ interprocessor times. The front-end-to-processor communication is separate. Additionally, the component tasks of the algorithm have been timed, namely the coefficient computations (TcoeÂ¡Â¡), the solver (TsoÂ¡ve), and the velocity-correction and convergence-checking parts. 3.3.1 Efficiency of Point and Line Solvers for the Inner Iterations Figure 3.2, based on timings made on the CM-5, illustrates the difference in parallel efficiency for SIMPLE using point-Jacobi and line-Jacobi iterative solvers. E is computed from Eq. 3.3 by timing Tcornm and Tcomp introduced above. Problem size is given in terms of the virtual processor ratio VP previously defined. There are two implementations each with different data layouts, for point-Jacobi iteration. One ignores the distinction between virtual processors which are on the same physical processor and those which are on different physical processors. Each array element is treated as if it is a processor. Thus, interprocessor communication is generated whenever data is to be moved, even if the two virtual processors doÂ¬ ing the communication happen to be on the same physical processor. To be more precise, a call to the run-time communication library is generated for every array elÂ¬ ement. Then, those array elements (virtual processors) which actually reside on the same physical processor are identified and the communication is done as a memory operationâ€”but the unnecessary overhead of calling the library is incurred. Obviously there is an inefficiency associated with pretending that there are as many processors as array elements, but the tradeoff is that this is the most straightforward, and indeed the intended, way to do the programming. In Figure 3.2, this approach is labelled â€œNEWS,â€ with the symbol â€œo.â€ The other implementation is labelled â€œon-VU,â€ with 63 the symbol â€œ+,â€ to indicate that interprocessor communication between virtual proÂ¬ cessors on the same physical processor is being eliminatedâ€”the programming is in a sense being done â€œon-VU.â€ To indicate to the compiler the different layouts of the data which are needed, the programmer inserts compiler directives. For the â€œNEWSâ€ version, the arrays are laid out as shown in this example for a lk x lk grid and an 8 x 16 processor layout on the CM-5: REAL*8 A( 1024,1024) $CMF LAYOUT A(:BLOCK=128 :PROCS=8, :BLOCK=64 :PROCS=16) Thus, the subgrid shape is 128 x 64, with a subgrid size {VP) of 8192 (this hapÂ¬ pens to be the biggest problem size for my program on a 32-node CM-5 with 4GBytes of memory). When shifting all the data to their east nearest-neighbor, for example, by far the large majority of transfers are on-VU and could be done without real interÂ¬ processor communication. But there are only 2 dimensions in A, so that data-parallel program statements cannot specifically access certain array elements, i.e. the ones on the perimeter of the subgrid. Thus it is not possible with the â€œNEWSâ€ layout to treat interior virtual processors differently from those on the perimeter, and conseÂ¬ quently data shifts between the interior virtual processors generate interprocessor communication even though it is unnecessary. In the â€œon-VUâ€ version, a different data layout is used which makes explicit to the compiler the boundary between physical processors. The arrays are laid out without virtual processors: $CMF LAYOUT A(:SERIAL,:SERIAL,:BLOCK=l :PROCS=8,:BLOCK=l :PROCS=16) The declaration must be changed accordingly, to A(128,64,8,16). Normally it is inconvenient to work with the arrays in this manner. Thus the approach taken here 64 is to use an â€œarray aliasâ€ of A [84]. In other words, this is an EQUIVALENCE funcÂ¬ tion for the data-parallel arrays (similar to the Fortran77 EQUIVALENCE concept), which equates A( 1024,1024) with A( 128,64,8,16), with the different LAYOUTs given above. It is the alias instead of the original A which is used in the on-VU point- Jacobi implementation. In the solver, the â€œon-VUâ€ layout is used; everywhere else, the more convenient â€œNEWSâ€ layout is used. The actual mechanism by which the equivalencing of distributed arrays can be accomplished is not too difficult to underÂ¬ stand. The front-end computer stores â€œarray descriptors,â€ which contain the array layout, the starting address in processor memory, and other information. The actual layout in each processorsâ€™ memory is linear and doesnâ€™t change, but multiple array descriptors can be generated for the same data. This descriptor multiplicity is what array aliasing accomplishes. With the â€œon-VUâ€ programming style, the compiler does not generate communication when the shift of data is along a SERIAL axis. Thus, interprocessor communication is generated only when the virtual processors involved are on different physical processors, i.e. only when it is truly necessary. The difference in the amount of communication is substantial for large subgrid sizes. For both the â€œNEWSâ€ and the â€œon-VUâ€ curves in Figure 3.2, E is initially very low, but as VP increases, E rises until it reaches a peak value of about 0.8 for the â€œNEWSâ€ version and 0.85 for the â€œon-VUâ€ version. The trend is due to the amorÂ¬ tization of the front-end-to-processor and off-VU (between VUs which are physically under control of different SPARC nodes) communication. The former contributes a constant overhead cost per Jacobi iteration to Tcomm, while the latter has a VP1/2 dependency [3]. However, it does not appear from Figure 3.2 that these two termsâ€™ effects can be distinguished from one another. For VP > 2k, the CM-5 is computing roughly 3/4 of the time for the implementaÂ¬ tion which uses the â€œNEWSâ€ version of point-Jacobi, with the remainder split evenly 65 between front-end-to-processor communication and on-VU interprocessor communiÂ¬ cation. It appears that the â€œon-VUâ€ version has more front-end-to-processor comÂ¬ munication per iteration, so there is, in effect, a price of more front-end-to-processor communication to pay in exchange for less interprocessor communication. ConseÂ¬ quently it takes VP > 4k to reach peak efficiency instead of 2k with the â€œNEWSâ€ version. For VP > 4k, however, E is about 5% ~ 10% higher than for the â€œNEWSâ€ version because the on-VU communication has been replaced by straight memory operations. The observed difference would be even greater if a larger part of the total parallel run time was spent in the solver. For the large VP cases in Figure 3.2, approximately equal time was spent computing coefficients and solving the systems of equations. â€œTypicalâ€ numbers of inner iterations were used, 3 each for the u and v momentum equations, and 9 for the p' equation. From Figure 3.2, then, it appears that the adÂ¬ vantage of the â€œon-VUâ€ version over the â€œNEWSâ€ version of point-Jacobi relaxation within the SIMPLE algorithm is around 0.1 in E, for large problem sizes. Red/black analogues to the â€œNEWSâ€ and â€œon-VUâ€ versions of point-Jacobi iterÂ¬ ation have also been tested. Red/black point iteration done in the â€œon-VUâ€ manner does not generate any additional front-end-to-processor communication, and thereÂ¬ fore takes almost an identical amount of time as point-Jacobi. Thus red/black point iterations are recommended when the â€œon-VUâ€ layout is used due to their improved convergence rate. However, with the â€œNEWSâ€ layout, red/black point iteration genÂ¬ erates two code blocks instead of one, and reduces by 2 the amount of computation per code block. This results in a substantial (~ 35% for the VP = 8k case) inÂ¬ crease in run time. Thus, if using â€œNEWSâ€ layouts, red/black point iteration is not cost-effective. 66 There are also two implementations of line-Jacobi iteration. In both, one inner iteration consists of forming a tridiagonal system of equations for the unknowns in each vertical line by moving the east/west terms to the right-hand side, solving the multiple systems of equations simultaneously, and repeating the procedure for the horizontal lines. In the first version, parallel cyclic reduction is used to solve the multiple tridiagÂ¬ onal systems of equations (see [44] for a clear presentation). This involves combining equations to decouple the system into even and odd equations. The result is two tridiagonal systems of equations each half the size of the original. The reduction step is repeated log2 N times, where N is the number of unknowns in each line. Thus, the computational operation count is 0(Nlog2N). Interprocessor communication occurs for every unknown for every step, thus the communication operation count is also 0(Nlog2N). However, the distance for communication increases every step of the reÂ¬ duction by a factor of 2. For the first step, nearest-neighbor communication occurs, while for the second step, the distance is 2, then 4, etc. Thus, the net communiÂ¬ cation speed is slower than the nearest-neighbor type of communication. Figure 3.2 confirms this argumentâ€”E peaks at about 0.5 compared to 0.8 for point-Jacobi itÂ¬ eration. In other words, for VP > 4k, interprocessor communication takes as much time as computation with the line-Jacobi solver using cyclic reduction. In the second version, the multiple systems of tridiagonal equations are solved using the standard TDMA algorithm along the lines. To implement this version, one must remap the arrays from (:NEWS,:NEWS) to (:NEWS,:SERIAL), for the vertical lines, and to (:SERIAL,:NEWS) for the horizontal lines. This change from rectangular subgrids to 1-d slices is the most time-consuming step, involving a global communication of data (â€œSENDâ€ instead of â€œNEWSâ€). Applied along the serial diÂ¬ mension, the TDMA does not generate any interprocessor communication. Some 67 front-end-to-processor communication is generated by the incrementing of the D0- loop index, but unrolling the DO-loop helps to amortize this overhead cost to some extent. Thus, in Figure 3.2 E is approximately constant at 0.14, except for very small VP. The global communication is much slower than computation and consequently there is not enough computation to amortize the communication. Furthermore, the constant E implies from Eq. 3.3 that Tcomm and Tcomp both scale in the same way with problem size. It is evident that Tcomp ~ VP because the TDM A is O(N). Thus constant E implies Tcomm ~ VP. This means doubling VP doubles Tcomm, indicating the communication speed has reached its peak, which further indicates that the full bandwidth of the fat-tree is being utilized. The disappointing performance of the standard line-iterative approach using the TDMA points out the important fact that, for the CM-5, global communication within inner iterations is intolerable. There is not enough computation to amortize slow communication in the solver for any problem size. With parallel cyclic reduction, where the regularity of the data movement allows faster communication, the efficiency is much higher, although still significantly lower than for point-iterations. Additional improvement can be sought by using the â€œon-VUâ€ data layout to implement the line-iterative solver within each processorâ€™s subgrid. This implementation essentially trades interprocessor communication for the front-end-to-PE type of communication, and in practice a front-end bottleneck develops. For the remainder of the discussion, all line-Jacobi results refer to the parallel cyclic reduction implementation. On the MP-1, the front-end-to-processor communication is not a major concern, as can be inferred from Figure 3.3. The efficiency of the SIMPLE algorithm using the point-Jacobi solver is plotted for each machine for the range of problem sizes corresponding to the cases solved on the MP-1. The CM-2 and CM-5 can solve much larger problems, so for comparison purposes only part of their data is shown. 68 Also, because the computers have different numbers of processors, the number of grid points is used instead of VP to define the problem size. As in Figure 3.2, each curve exhibits an initial rise corresponding to the amortizaÂ¬ tion of the front-end-to-processor communication and, for the CM-2 and CM-5, the off-processor â€œNEWSâ€ communication. On the MP-1, peak E is reached for small problems (VP > 32). Due to the MP-lâ€™s relatively slow processors, the computaÂ¬ tion time quickly amortizes the front-end-to-processor communication time as VP increases. Furthermore, because the relative speed of X-Net communication is fast, the peak E is high, 0.85. On the CM-2, the peak E is 0.4, and this efficiency is reached for approximately VP > 128. On the CM-5, the peak E is 0.8, but this efficiency is not reached until VP > 2k. If computation is fast, then the rate of inÂ¬ crease of E with VP depends on the relative cost of on-processor, off-processor, and front-end-to-processor communication. If the on-processor communication is fast, larger VP is required to reach peak E. Thus, on the CM-5, the relatively fast on-VU communication is simultaneously responsible for the good (0.8) peak E, and the fact that very large problem sizes, (VP > 2k, 64 times larger than on the MP-1), are needed to reach this peak E. The aspect ratio of the virtual subgrid constitutes a secondary effect of the data layout on the efficiency for hierarchical mapping. The major influence on E depends on VP, i.e. the subgrid size, but the subgrid shape matters, too. This dependence comes into play due to the different speeds of the on-processor and off-processor types of communication. Higher aspect ratio subgrids have higher area to perimeter ratios, and thus relatively more of off-processor communication than square subgrids. Figure 3.4 gives some idea of the relative importance of the subgrid aspect ratio effect. Along each curve the number of grid points is fixed, but the grid dimensions vary, which, for a given processor layout, causes the subgrid shape (aspect ratio), to 69 vary. For example, on the CM-5 with an 8 x 16 processor layout, the following grids were used corresponding to the VP = 1024 CM-5 curve: 256 x 512, 512 x 256, 680 x 192, and 1024 x 128. These cases give subgrid aspect ratios of 1, 4, 7, and 16. Tnews is the time spent in â€œNEWSâ€ type of interprocessor communication and Tcomp is the time spent doing computation during 100 SIMPLE iterations. The solver for these results is point-Jacobi relaxation. For the VP â€” 1024 CM-5 case, increasing the aspect ratio from 1 to 16 causes Tnews/Tcomp to increase from 0.3 to 0.5. This increase in Tnews/Tcomp increases the run time for 100 iterations from 15s to 20s, and decreases the efficiency from 0.61 to 0.54. For the VP = 8192 CM-5 case, increasing the aspect ratio from 1 to 16 causes TnewslTcomp to increase from 0.19 to 0.27. This increase in Tnews/Tcomp increases the run time for 100 iterations from 118s to 126s, and decreases the efficiency from 0.74 to 0.72. Thus, the aspect ratio effect diminishes as VP increases due to the increasing area of the subgrid. In other words the variation in the perimeter length matters less, percentage-wise, as the area increases. The CM-2 results are similar. However, on the CM-2 the on-PE type of communication is slower than on the CM-5, relative to the computational speed. Thus, Tnews/Tcomp ratios are higher on the CM-2. 3.3.2 Effect of Uniform Boundary Condition Implementation In addition to the choice of solver, the treatment of boundary coefficient computaÂ¬ tions was discussed earlier as an important consideration affecting parallel efficiency. Figure 3.5 compares the implementation described in the introductory section of this chapter, to an implementation which treats the boundary control volumes separate from the interior control volumes. The latter approach involves some 1-d operations which leave some processors idle. 70 The results indicated in Figure 3.5 were obtained on the CM-2, using point- Jacobi relaxation as the solver. With the uniform approach, the ratio of the time spent computing coefficients, Tcoejf, to the time spent solving the equations, Tsoive, remains constant at 0.6 for VP > 256. Both TcoeÂ¡Â¡ and Tsoive ~ VP in this case, so doubling VP doubles both TcoeÂ¡j and Tsoive, leaving their ratio unchanged. The value 0.6 reflects the relative cost of coefficient computations compared to point-Jacobi iteration. There are three equations for which coefficients are computed and 15 total inner iterations, 3 each for the u and v equations, and 9 for the p' equation. Thus if more inner iterations are taken, the ratio of Tcoejj to TsoÂ¡ve will decrease, and vice- versa. With the 1-d implementation, Tcoefj/TsoÂ¡ve increases until VP > 1024. Both Tcoeff and Tsoive scale with VP asymptotically, but Figure 3.5 shows that Tcoejj has an apparently very significant square-root component due to the boundary operations. If N is the number of grid points and nv is the number of processors, then VP = N/np. For boundary operations, N1^2 control volumes are computed in parallel with only nxJ2 processorsâ€”hence the VPcontribution to Tcoejj. From Figure 3.5, it appears that very large problems are required to reach the point where the interior coefficient computations amortize the boundary coefficient computations. Even for large VP when Tcoefj/Tsoive is approaching a constant, this constant is larger, approximately 0.8 compared to 0.6 for the uniform approach, due to the additional front-end-to- processor communication which is intrinsic to the 1-d formulation. 3.3.3 Overall Performance Table 3.1 summarizes the relative performance of SIMPLE on the CM-2, CM-5, and MP-1 computers, using point and line-iterative solvers and the uniform boundary 71 condition treatment. In the first three cases the â€œNEWSâ€ implementation of point- Jacobi relaxation is the solver, while the last two cases are for the line-Jacobi solver using cyclic reduction. Machine Solver Problem Size VP T 1 V Time/Iter./Pt. Speed ) (MFlops) Peak Speed 512 PE CM-2 Point- Jacobi 512 x 1024 1024 188 s 2.6 x 10~6 s 147 4 128 VU CM-5 Point- Jacobi 736 x 1472 8192 137 s 1.3 x 10â€œ6 s 417 10 1024 PE MP-1 Point- Jacobi 512 x 512 256 316 s 1.2 x 10~5 s 44* 59 512 PE CM-2 Line- J acobi 512 x 1024 1024 409 s 7.8 x 10"6 s 133 3 128 VU CM-5 Line- Jacobi 736 x 1472 8192 453 s 4.2 x 10"6 s 247 6 Table 3.1. Performance results for the SIMPLE algorithm for 100 iterations of the model problem. The solvers are the point-Jacobi (â€œNEWSâ€) and line-Jacobi (cyclic reduction) implementations. 3, 3, and 9 inner iterations are used for the u, v, and pâ€™ equations, respectively. * The speeds are for double-precision calculations, except on the MP-1. In Table 3.1, the speeds reported are obtained by comparing the timings with the identical code timed on a Cray C90, using the Cray hardware performance monÂ¬ itor to determine Mflops. In terms of Mflops, the CM-2 version of the SIMPLE algorithmâ€™s performance appears to be consistent with other CFD algorithms on the CM-2. Jesperson and Levit [44] report 117 Mflops for a scalar implicit version of an approximate factorization Navier-Stokes algorithm using parallel cyclic reduction to solve the tridiagonal systems of equations. This result was obtained for a 512 x 512 simulation of 2-d flow over a cylinder using a 16k CM-2 as in the present study (a different execution model was used (see [3, 47] for details). The measured time per time-step per grid point was 1.6 x 10-5 seconds. By comparison, the performance of the SIMPLE algorithm for the 512 x 1024 problem size using the line-Jacobi solver is 72 133 Mflops and 7.8 x 10-6 seconds per iteration per grid pt. Egolf [20] reports that the TEACH Navier-Stokes combustor code based on a sequential pressure-based method with a solver that is comparable to point-Jacobi relaxation, obtains a performance which is 3.67 times better than a vectorized Cray X-MP version of the code, for a model problem with 3.2 x 104 nodes. The present program runs 1.6 times faster than a single Cray C90 processor for a 128 x 256 problem (32k grid points). One Cray C-90 processor is about 2-4 times faster than a Cray X-MP. Thus, the present code runs comparably fast. 3.3.4 Isoefficiencv Plot Figures 3.2-3.4 addressed the effects of the inner-iterative solver, the boundary treatment, the data layout, and the variation of parallel efficiency with problem size for a fixed number of processors. Varying the number of processors is also of interest and, as discussed in Chapter 1, an even more practical numerical experiment is to vary np in proportion with the problem size, i.e. the scaled-size model. Figure 3.6, which is based on the point-Jacobi MP-1 timings, incorporates the above information into one plot, which has been called an isoefficiency plot by Kumar and Singh [46]. The lines are paths along which the parallel efficiency E remains constant as the problem size and the number of processors np vary. Using the point- Jacobi solver and the uniform boundary coefficient implementation, each SÃMPLE iteration has no substantial contribution from operations which are less than fully parallel or from operations whose time depends on the number of processors. The efficiency is only a function of the virtual processor ratio, thus the lines are straight. Much of the parameter space is covered by efficiencies between 0.6 and 0.8. The reason that the present implementation is linearly scalable is that the operÂ¬ ations are all scalableâ€”each StMPLE iteration has predominantly nearest-neighbor 73 communication and computation and full parallelism. Thus, Tp depends on VP. Local communication speed does not depend on np. T\ depends on the problem size N. Thus, as N and np are increased in proportion, starting from some initial ratio, the efficiency from Eq. 3.3 stays constant. If the initial problem size is large and the corresponding parallel run time is acceptable, then one can quickly get to very large problem sizes while still maintaining Tp constant by increasing np a relatively small amount (along the E = 0.85 curve). If the desired run time is smaller, then initially (i.e. starting from small np) the efficiency will be lower. Then the scaled-size experiment requires relatively more processors to get to a large problem size along the constant efficiency (constant Tp for point-Jacobi ierations) curve. Thus, the most desirable situation occurs when the efficiency is high for an initially small problem size. For this case the fixed-time and scaled-size methods are equivalent, because the problem size T\ depends on N per iteration. However this is not the case when the SIMPLE inner iterations are done with the line-Jacobi solver using parallel cyclic reduction. Cyclic reduction requires (131og2 N + l)N operations to solve a tridiagonal system of N equations [44]. Thus, T\ ~ (131og2 N + l)N and on np â€” N processors, Tp ~ 13 log2 N 1 because every processor is active during every step of the reduction and there are 13 log2 N-(-1 steps. Since VP = 1, every processorâ€™s time is proportional to the number of steps, assuming each step costs about the same. In the scaled-size approach, one doubles np and N together, which therefore gives Ti ~ (261og2 2N-\-2)N and Tp ~ 13 log2 2N+1. The efficiency is 1, but Tp is increased and 7\ is more than doubled. In the fixed-time approach, then, one concludes that N must be increased by a factor which is less than two, and np must be doubled, in order to maintain constant Tp. If a plot like Figure 3.6 is constructed, it should be done with T\ instead of N as the measure of problem size. In that case, the lines 74 of constant efficiency would be described as T\ ~ npi with a > 1. The ideal case is a = 1. In addition to the operation count, there is another factor which reduces the scalability of cyclic reduction, namely the time per step is not actually the same as was assumed aboveâ€”later steps require communication over longer distances which is slower. In practice, however, no more than a few steps are necessary because the coupling between widely-separated equations becomes very weak. As the system is reduced the diagonal becomes much larger than the off-diagonal terms which can then be neglected and the reduction process abbreviated. In short, the basic prerequisite for scaled-size constant efficiency is that the amount of work per SIMPLE iteration varies with VP and that the overheads and inefficiencies, specifically the time spent in communication and the fraction of idle processors, do not grow relative to the useful computational work as np and N are increased proportionally. The SIMPLE implementation developed here using the point-iterative solvers, Jacobi and red/black, have this linear computational scalabilÂ¬ ity property. On the other hand, the convergence rate of point-iterative methods increases at a rate greater than the problem size, so although Tp can be maintained constant while the problem size and np are scaled up, the convergence rate deteriorates. Hence the total run time (cost per iteration multiplied by the number of iterations) increases. This lack of numerical scalability of standard iterative methods like point-Jacobi relaxation is the motivation for the development of multigrid strategies. 3.4 Concluding Remarks The SIMPLE algorithm, especially using point-iterative methods, is efficient on SIMD machines and can maintain a relatively high efficiency as the problem size and the number of processors is scaled up. However, boundary coefficient computations 75 need to be folded in with interior coefficient computations to achieve good efficiencies at smaller problem sizes. For the CM-5, the inefficiency caused by idle processors in a 1-d boundary treatment was significant over the entire range of problem sizes tested. The line-Jacobi solver based on parallel cyclic reduction leads to a lower peak E (0.5 on the CM-5) than the point-Jacobi solver (0.8), because there is more communication and on average this communication is less localized. On the other hand, the asymptotic convergence rates of the two methods are also different and need to be considered on a problem-by-problem basis. The speeds which are obtained with the line-iterative method are consistent and comparable with other CFD algorithms on S1MD computers. The key factor in obtaining high parallel efficiency for the StMPLE algorithm on the computers used, is fast nearest-neighbor communication relative to the speed of computation. On the CM-2 and CM-5, hierarchical mapping allows on-processor communication to dominate the slower off-processor form(s) of communication for large VP. The efficiency is low for small problems because of the relatively large contribution to the run time from the front-end-to-processor type of communication, but this type of communicaton is constant and becomes less important as the problem size increases. Once the peak E is reached, the efficiency is determined by the balance of compuÂ¬ tation and on-processor communication speedsâ€”for the CM-5, using a point-Jacobi solver, E approaches approximately 0.8, while on the CM-2 the peak efficiency is 0.4, which reflects the fact that the CM-5 vector units have a better balance, at least for the operations in this algorithm, than the CM-2 processors. The rate at which E approaches the peak value depends on the relative contribuÂ¬ tions of on- and off-processor communication and front-end-to-processor communicaÂ¬ tion to the total run time. On the CM-5, VP > 2k is required to reach peak E. This 76 problem size is about one-fourth the maximum size which can be accommodated, and yet still larger than many computations on traditional vector supercomputers. Clearly a gap is developing between the size of problems which can be solved effiÂ¬ ciently in parallel and the size of problems which are small enough to be solved on serial computers. For parallel computations of all but the largest problems, then, the data layout issue is very important- in going from a square subgrid to one with aspect ratio of 16, for a VP = lk case on the CM-5, the run time increased by 25%. On the MP-1, hierarchical mapping is not needed, because the processors are slow compared to the X-Net communication speed. The peak E is 0.85 with the point-Jacobi solver, and this performance is obtained for VP > 32, which is about one-eighth the size of the largest case possible for this machine. Thus, with regards to achieving efficient performance in the teraflops range, the comparison given here suggests a preference for numerous slow processors instead of fewer fast ones, but such a computer may be difficult and expensive to build. 77 4 x 1 Layout of Processors PEO PE 1 PE 2 PE 3 Array A(8) Cut-and-Stack Mapping (MP-Fortran) Hierarchical Mapping (CM-Fortran) Memory Layers A ^ HZ ^ 8^ PE 0 PE 1 PE 2 PE 3 PE 0 PE 1 PE 2 PE 3 2x1 virtual subgrids i â€¢ 2 3 â€¢ 4 5 i 6 7 â€¢ s Figure 3.1. Mapping an 8 element array A onto 4 processors. For the cut-and- stack mapping, nearest-neighbors array elements are mapped to nearest-neighbor physical processors. For the hierarchical mapping, nearest-neighbor array elements are mapped to nearest-neighbor virtual processors, which may be on the same physical processor. 78 1 0.8 0.6 LD 0.4 0.2 0 0 5000 10000 VP Efficiency vs. VP Â© Â© 9 (T>g/ ' x Â¡fe>*x S x x + o + o * x X + o X + XX Point-Jacobi (on-VU) Point-Jacobi (NEWS) Line-Jacobi (Cyclic Red.) Line-Jacobi (TDMA) X XX Figure 3.2. Parallel efficiency, i?, as a function of problem size and solver, for the CM-5 cases. The number of grid points is the virtual processor ratio, VP, multiplied by the number of processors, 128. E is computed from Eq. 3.3. It reflects the relative amount of communication, compared to computation, in the algorithm. 79 E vs. Problem Size Figure 3.3. Comparison between the CM-2, CM-5 and MP-1. The variation of parallel efficiency with problem size is shown for the model problem, using point- Jacobi relaxation as the solver. E is calculated from Eq. 3.3, and 7\ = npTcomp for the CM-2 and CM-5, where Tcomp is measured. For the MP-1 cases. T\ is the front-end time, scaled down to the estimated speed of the MP-1 processors (0.05 Mflops). Tnews/Tcomp 80 Aspect Ratio Effect Subgrid AR Figure 3.4. Effect of subgrid aspect ratio on interprocessor communication time, Tnews, for the hierarchical data-mapping (CM-2 and CM-5). Tnews is normalized by Tcomp In order to show how the aspect ratio effect varies with problem size, without the complication of the fact that Tcomp varies also. Tcoeff/Tsolve 81 Effect of Implementation VP Figure 3.5. Normalized coefficient computation time as a function of problem size, for two implementations (on the CM-2). In the 1-d case the boundary coefficients are handled by 1-d array operations. In the 2-d case the uniform implementaton computes both boundary and interior coefficients simultaneously. TcoeÂ¡Â¡ is the time spent computing coefficients in a SIMPLE iteration; TsoÂ¡ve is the time spent in point- Jacobi iterations. There are 15 point-Jacobi iterations (i/u = vv = 3 and uc = 9). 82 Isoefficiency Curves i ... I I i 2000 4000 6000 8000 # Processors (MP-1) Figure 3.6. Isoefficiency curves based on the MP-1 cases and SIMPLE method with the point-Jacobi solver. Efficiency E is computed from Eq. 3.3. Along lines of constant E the cost per SIMPLE iteration is constant with the point-Jacobi solver and the uniform boundary condition implementation. CHAPTER 4 A NONLINEAR PRESSURE-CORRECTION MULTIGRID METHOD The single-grid timing results focused on the cost per iteration in order to elucidate the computational issues which influence the parallel run time and the scalability. But the parallel run time is the cost per iteration multiplied by the number of iterations. For scaling to large problem sizes and numbers of processors, the numerical method must scale well with respect to convergence rate, also. The convergence rate of the single-grid pressure-correction method deteriorates with increasing problem size. This trait is inherited from the smoothing property of the stationary linear iterative method, point or line-Jacobi relaxation, used to solve the systems of u, v, and p' equations during the course of SIMPLE iterations. Point- Jacobi relaxation requires 0(N2) iterations, where N is the number of grid points, to decrease the solution error by a specified amount [1], In other words, the number of iterations increases faster than the problem size. At best the cost per iteration stays constant as the number of processors np increases proportional to the problem size. Thus, the total run time increases in the scaled-size experiment using single-grid pressure-correction methods, due to the increased number of iterations required. This lack of numerical scalability is a serious disadvantage for parallel implementations, since the target problem size for parallel computation is very large. Multigrid methods can maintain good convergence rates as the problem size inÂ¬ creases. For Poisson equations, problem-size independent convergence rates can be obtained [36, 55]. The recent book by Briggs [10] introduces the major concepts in 83 84 the context of Poisson equations. See also [11, 37, 90] for surveys and analyses of multigrid convergence properties for more general linear equations. For a description of practical techniques and special considerations for fluid dynamics, see the imporÂ¬ tant early papers by Brandt [5, 6]. However, there are many unresolved issues for application to the incompressible Navier-Stokes equations, especially with regards to their implementation and performance on parallel computers. The purpose of this chapter is to describe the relevant convergence rate and stability issues for multigrid methods in the context of application to the incompressible Navier-Stokes equations, with numerical experiments used to illustrate the points made, in particular, regardÂ¬ ing the role of the restriction and prolongation procedures. 4.1 Background The basic concept is the use of coarse grids to accelerate the asymptotic conÂ¬ vergence rate of an inner iterative scheme. The inner iterative method is called the â€œsmootherâ€ for reasons to be made clear shortly. In the context of the present applicaÂ¬ tion to the incompressible Navier-Stokes equations, the single-grid pressure-correction method is the inner iterative scheme. Because the pressure-correction algorithm also uses inner iterationsâ€”to solve the systems of u, v, and p' equationsâ€”the multigrid method developed here actually has three nested levels of iterations. A multigrid V cycle begins with a certain number of smoothing iterations on the fine grid, where the solution is desired. Figure 4.1 shows a schematic of a V(3,2) cycle. In this case three pressure-correction iterations are done first. Then residuals and variables are restricted (averaged) to obtain coarse-grid values for these quantities. The solution to the coarse-grid discretized equation provides a correction to the fine- grid solution. Once the solution on the coarse grid is obtained, the correction is interpolated (prolongated) to the fine grid and added back into the solution there. 85 Some post-smoothing iterations, two in this case, are needed to eliminate errors introduced by the interpolation. Since it is usually too costly to attempt a direct solution on the coarse grid, this smoothing-correction cycle is applied recursively, leading to the V cycle shown. The next section describes how such a procedure can accelerate the convergence rate of an iterative method, in the context of linear equations. The multigrid scheme for nonlinear scalar equations and the Navier-Stokes system of equations is then described. Brandt [5] was the first to formalize the manner in which coarse grids could be used as a convergence-acceleration technique for a given smoother. The idea of using coarse grids to generate initial guesses for fine-grid solutions was around much earlier. The cost of the multigrid algorithm, per cycle, is dominated by the smoothing cost, as will be shown in Chapter 5. Thus, with regard to the parallel run time per multigrid iteration, the smoother is the primary concern. Also, with regard to the convergence rate, the smoother is important. The single-grid convergence rate characteristics of pressure-correction methods, the dependence on Reynolds number, flow problem, and the convection scheme, carry over to the multigrid context. However, in the multigrid method the smootherâ€™s role is, as the name implies, to smooth the fine-grid residual, which is a different objective than to solve the equations quickly. A smooth fine-grid residual equation can be approximated accurately on a coarser grid. The next section describes an alternate pressure-based smoother, and compares its cost against the pressure-correction method on the CM-5. Stability of multigrid iterations is also an important unresolved issue. There are two ways in which multigrid iterations can be caused to diverge. First, the single-grid smoothing iterations can diverge, for example if central-differencing is used there are possibly stability problems if the Reynolds number is high. Second, poor coarse-grid 86 corrections can cause divergence if the smoothing is insufficient. In a sense this latter issue, the scheme and intergrid transfer operators which prescribe the coordination between coarse and fine grids in the multigrid procedure, is the key issue. In the next section two â€œstabilization strategiesâ€ are described. Then, the impact of different restriction and prolongation procedures on the convergence rate is studied in the context of two model problems, lid-driven cavity flow and flow past a symmetric backward-facing step. These two particular flow problems have different physical characteristics, and therefore the numerical experiments should give insight into the problem-dependence of the results. 4.1.1 Terminology and Scheme for Linear Equations The discrete problem to be solved can be written Ahuh = Sh, corresponding to some differential equation L[u] â€” S. The set of values uh is defined by K',j} = u{ih,jh), (i,j) e ([0 : N], [0 : N]) = Ãœh. (4.1) Similarly, u2h is defined on the coarser grid Q2h with grid spacing 2h. The variable u can be a scalar or a vector, and the operator A can be linear or nonlinear. For linear equations, the â€œcorrection schemeâ€ (CS) is frequently used. A two- level multigrid cycle using CS accelerates the convergence of an iterative method (with iteration matrix P) by the following procedure: Do v fine-grid iterations vh 4â€” Pvvh Compute residual on flh rh = Ahvh â€” Sh Restrict rh to Q2h r2h = I2hrh Solve exactly for e2h e2h = Correct vh on fC (u/l)netÃ¼ = (vh)old + I%he2h 87 72/l and Â¡2h symbolize the restriction and prolongation procedures. The quantity vh is the current approximation to the discrete solution uh. The algebraic error is the difference between them, eh = uh â€” vh. The discretization error is the difference between the exact solutions of the continuous and discrete problems, e<Â¿tscr = u â€” uh. The truncation error is obtained by substituting the exact solution into the discrete equation, rh = Ahu -Sh = Ahu - Ahuh. (4.2) The notation above follows Briggs [10]. The two-level multigrid cycle begins on the fine grid with u iterations of the smoother. Standard iterative methods all have the â€œsmoothing property,â€ which is that the various eigenvector-decomposed components of the solution error are damped at a rate proportional to their corresponding eigenvalues, i.e. the high frequency errors are damped faster than the low frequency (smooth) errors. Thus, the converÂ¬ gence rate of the smoothing iterations is initially rapid, but deteriorates as smooth error components, those with large eigenvalues, dominate the remaining error. The purpose of transferring the problem to a coarser grid is to make these smooth error components appear more oscillatory with respect to the grid spacing, so that the initial rapid convergence rate is obtained for the elimination of these smooth errors by coarse-grid iterations. Since the coarse grid Q2h has only 1/4 as many grid points as Qh (in 2-d), the smoothing iterations on the coarse grid are cheaper as well as more effective in reducing the smooth error components than on the fine grid. In the correction scheme, the coarse-grid problem is an equation for the algebraic error, A^he2h _ y*2/l (4.3) 88 approximating the fine-grid residual equation for the algebraic error. To obtain the coarse-grid source term, r2/l, the restriction procedure I2h is applied to the fine-grid residual rh, r2h = I2hhrh. (4.4) Eq. 4.4 is an averaging type of operation. Two common restriction procedures are straight injection of fine-grid values to their corresponding coarse-grid grid points, and averaging rh over a few fine-grid grid points which are near the corresponding coarse-grid grid point. The initial error on the coarse grid is taken as zero. After the solution for e2h is obtained, this coarse-grid quantity is interpolated to the fine grid and used to correct the fine-grid solution, vh <- vh + I?he2h. (4.5) For I%hl common choices are bilinear or biquadratic interpolation. In practice the solution for e2h is obtained by recursion on the two-level cycleâ€” (A2h)~l is not explicitly computed. On the coarsest grid, direct solution may be feasible if the equation is simple enough. Otherwise a few smoothing iterations can be applied. Recursion on the two-level algorithm leads to a â€œV cycle,â€ as shown in Figure 4.1. A simple V(3,2) cycle is shown. Three smoothing iterations are taken before reÂ¬ stricting to the next coarser grid, and two iterations are taken after the solution has been corrected. The purpose of the latter smoothing iterations is to smooth out any high-frequency noise introduced by the prolongation. Other cycles can be enviÂ¬ sioned. In particular the W cycle is popular [6]. The cycling strategy is called the â€œgrid-schedule.â€ since it is the order in which the various grid levels are visited. The most important consideration for the correction scheme has been saved for last, namely the definition of the coarse-grid discrete equation A2h. One possibility is 89 to discretize the original differential equation directly on the coarse grid. However this choice is not always the best one. The convergence-rate benefit from the multigrid strategy is derived from the particular coarse-grid approximation to the fine-grid discrete problem, not the continuous problem. Because the coarse-grid solutions and residuals are obtained by particular averaging procedures, there is an implied averaging procedure for the fine-grid discrete operator Ah which should be honored to ensure a useful homogenization of the fine-grid residual equation. This issue is critical when the coefficients and/or dependent variables of the governing equations are not smooth [17]. For the Poisson equation, the Galerkin approximation A2h = I2hAhI%h is the right choice. The discretized equation coefficients on the coarse grid are obtained by applying suitable averaging and interpolation operations to the fine-grid coeffiÂ¬ cients, instead of by discretizing the governing equation on a grid with a coarser mesh spacing. Briggs has shown, by exploiting the algebraic relationship between bilinear interpolation and full-weighting restriction operators, that initially smooth errors begin in the range of interpolation and finish, after the smoothing-correction cycle is applied, in the null space of the restriction operator [10]. Thus, if the fine-grid smoothing eliminates all the high-frequency error components in the solution, one V cycle using the correction-scheme is a direct solver for the Poisson equation. The conÂ¬ vergence rate of multigrid methods using the Galerkin approximation is more difficult to analyze if the governing equations are more complicated than Poisson equations, but significant theoretical advantages for application to general linear problems have been indicated [90]. 90 4.1.2 Full-Approximation Storage Scheme for Nonlinear Equations The brief description given above does not bring out the complexities inherent in the application to nonlinear problems. There is only experience, derived mostly from numerical experiments, to guide the choice of the restriction/prolongation procedures and the smoother. Furthermore, the linkage between the grid levels requires special considerations because of the nonlinearity. The correction scheme using the Galerkin approximation can be applied to the nonlinear Navier-Stokes system of equations [94]. However, in order to use CS for nonlinear equations, linearization is required. The best coarse-grid correction only improves the fine-grid solution to the linearized equation. Also, for complex equaÂ¬ tions, considerable expense is incurred in computing A2h by the Galerkin approxiÂ¬ mation. The commonly adopted alternative is the intuitive one, to let A2h be the differential operator L discretized on the grid with spacing 2h instead of h. In exÂ¬ change for a straightforward problem definition on the coarse grid though, special restriction and prolongation procedures may be necessary to ensure the usefulness of the resulting corrections. Numerical experiments on a problem-by-problem basis are necessary to determine good choices for the restriction and prolongation procedures for Navier-Stokes multigrid methods. The full-approximation storage (FAS) scheme [5] is preferred over the correction scheme for nonlinear problems. The coarse-grid corrections generated by FAS improve the solution to the full nonlinear problem instead of just the linearized one. The discretized equation on the fine grid is, again, Ahuh = Sh. (4.6) 91 The approximate solution vh after a few fine-grid iterations defines the residual on the fine grid, Ahvk = Sh + rh. (4.7) A correction, the algebraic error e^lg = uh â€” vk, is sought which satisfies Â¿V + <4,) = S'1. (4.8) The residual equation is formed by subtracting Eq. 4.7 from Eq. 4.8, and cancelling Sk, Ah(vh + eh)-Ah{vh) = -rh, (4.9) where the subscript â€œalgâ€ is dropped for convenience. For linear equations the Ahvh terms cancel leaving Eq. 4.3. Eq. 4.9 does not simplify for nonlinear equations. Assuming that the smoother has done its job, rh is smooth and Eq. 4.9 is the same as the coarse-grid residual equation A2h{v2h + e2h) - A2h(v2h) = -r2h, (4.10) at coarse-grid grid points. The error e2h is to be found, interpolated back to Uh according to eh = and added to vh so that Eq. 4.8 is satisfied. The known quantities are v2h, which is a â€œsuitableâ€ restriction of vh, and r2h, likewise a restriction of rh. Different restrictions can be used for residuals and solutions. Thus, Eq. 4.10 can be written A2h(I2hvh + e2h) = A2h(I2hvh) - I2hhrh. (4.11) Since Eq. 4.11 is not an equation for e2h, one solves instead for the sum I2hvh + e2h. Expanding rh and regrouping terms, Eq. 4.11 can be written A2h(u2k) = A2h{I2hhvh) - I2hhrh (4.12) 92 = [A2h(I2hvh) - I2h(Ahvh) + I2hSh - S2/l] + 5 r n2h . q2h. numerical ' â€™ (4.13) (4.14) Eq. 4.14 is similar to Eq. 4.6 except for the extra numerically-derived source term. Once I2hvh + e2h is obtained the coarse-grid approximation to the fine-grid error, e2h, is computed by first subtracting the initial coarse-grid solution I2hvh, e2h = u- ,2h (4.15) then interpolating back to the fine grid and combining with the current solution, vh <- vh + 4(e2/i). (4.16) 4.1.3 Extension to the Navier-Stokes Equations The incompressible Navier-Stokes equations are a system of coupled, nonlinear equations. Consequently the FAS scheme given above for single nonlinear equations needs to be modified. The variables u\, u2, and Ug represent the cartesian velocity components and the pressure, respectively. Corresponding subscripts are used to identify each equationsâ€™ source term, residual and discrete operator in the formulation below. The three equations for momentum and mass conservation are treated as if part of the following matrix equation, â€™ A\ 0 Gy \<) r sÃ i o Ah2 Ghy u\ = si .Ghx Ghy 0 . . U3 . L si J The continuity equation source term is zero on the finest grid, Qh, but for coarser grid levels it may not be zero. Thus, for the sake of generality it is included in Eq. 4.17. Thus, for the iii-momentum equation Eq. 4.8 is modified to account for the pressure-gradient, G^u^i which is also an unknown. The approximate solutions are 93 v\, v%, and corresponding to u1}, uif, and u3. For the ui-momentum equation, the approximate solution satisfies A\vhx + Ghxv$ = S* + rf. (4.18) The fine-grid residual equation corresponding to Eq. 4.9 is modified to â– 4(4 + 4) - 4(4) + <4(4 + 4) - <4(4) = -4. (4.19) which is approximated on the coarse grid by the corresponding coarse-grid residual equation, A2h/2k 1 â€ž2h\ 4 2/i/â€ž,2/i\ . /-i2h 1 _.2h < __2/i\ /~i2h/ ,2h\ 2h /a <)rv\ A1 +el ) ~ Al 4l ) + (U3 + e3 ) ~ Gx (U3 ) - ~rl (4â€™2U) The known terms are v\h â€” v2h = I2hv^ and r\h = I2hrx. Expanding r\ and regrouping terms, Eq. 4.19 can be written A? Â«) + G2xh (uf) = Alh (/f v}) + Glh (/Â«) - r a 2h/ r2h.h\ , si2h/r2h^h\ = iAi (4 v\) + gx (4 v3) -llh(Axvx + Ghxv$) + /fS'f - 5^] + (4.21) = K numerical + Sf 2 h Since Eq. 4.22 includes numerically derived source terms in addition to the physical ones, the coarse-grid variables are not in general the same as would be obtained from a discretization of the original continuous governing equations on the coarse grid. The u2-momentum equation is treated similarly, and the coarse-grid continuity equation is Gf ,2 h + G\ 2hulh = Gl\llhu\) 1 /^i2hi r2hâ€ž,h\ r2/iâ€ž + Gy (4 U2) - 4 V 2hh (4.22) 94 The system of equations Eq. 4.17 are solved by either the pressure-correction method (sequential) or the locally-coupled explicit method described in the next section. In addition to the choice of the smoother, the specification of the coarse-grid discrete problem (A2h) is critical to the convergence rate, and to the stability of the multigrid iterations as well. In the description of the FAS scheme for the 2-d incompressible Navier-Stokes equations presented earlier, no mention was made of the coarse grid discretization. Intuitively, one would use the same discretization for each of the terms as on the fine grid. For example, if the convection terms are central- differenced on the fine grid, then central-differencing should be used on the coarse grid, also. However, with such an approach numerical stability frequently becomes a problem, particularly in high Reynolds number flow problems. 4.2 Comparison of Pressure-Based Smoothers The single-grid convergence rate of pressure-correction methods for the incomÂ¬ pressible Navier-Stokes equations depends strongly on the discretization of the nonÂ¬ linear convection terms, the Reynolds number, and the importance of the pressure- velocity coupling in the fluid dynamics. The grid size and quality can also affect the convergence rate in curvilinear formulations. These issues carry over to the multigrid context and are complicated by the interplay between the evolving solutions on the multiple grid levels. Two pressure-based methods are popular smoothers. The first is the pressure- correction method studied in Chapter 2 and 3, and the other is Vankaâ€™s locally- coupled explicit method [89] briefly introduced in Chapter 1. Much attention has been focused on comparing the performance of these two methods in the multigrid 95 context, i.e. as smoothers. The semi-implicit pressure-correction methods, due to their implicitness, are better single-grid solvers. In the locally-coupled explicit method, pressure and velocity are updated in a coupled manner instead of sequentially. A finite-volume implementation on a stagÂ¬ gered grid is employed. The pressure and the velocities on the faces of each p control volume are updated simultaneously. However the simultaneous update of pressure and velocity is only for one control volume at a time. Underrelaxation is again necessary due to the decoupling between control volumes. The control volumes are traversed by the lexicographical ordering with the most recently updated u and v values used when available. Thus the original method is called BGS (for â€œblock Gauss-Seidelâ€). After one sweep of the grid each u and v have been updated twice and each pressure once. A red-black ordering suitable for parallel computation has been developed in this research. By analogy, this algorithm is called BRB (block red-black). For the (i,j)th pressure control volume, the continuity equation is written in terms of the velocity corrections needed to restore mass conservation: \u t+l,j â– u'i,j)Ay+(vij+i-vi,j)Ax = Â«,â€¢â– "i+hj i)Ay+(vi,j-vlj+i)Ax = Ri,ji (4-23) where Rct ] is the mass residual in the (ij)th control volume. The notation follows the development in Chapter 2 except now that pressure and velocity are coupled it is necessary to refer to the (i,j) notation on occasion. In Figure 2.3, uw is iÂ¿Â¿j, ue is 1,j, rs is u1(j, and vn is c,,j-t-1 * The discrete u-momentum equation for the (i,j)th p control volume is written U f aPuij + Pi+uAy E k=E.W,N,S a,u kuk + (Pij - Pi+u)Ay - aPi = ~Rh (4.24) 96 The discretized momentum equations for the three other faces of the pressure control volume are written analogously, giving a system of five equations in five unknowns, (4.25) '(Â«;)Â« Ay r i -Ay â€œi+ij DU â€¢H-.+lj Ax = K)w â€” Ax
r>vâ€”Ay Ay â€”Ax Ax 0 L J The solution of this matrix equation is done by hand for p[j, _ (A(A_ (Ax)Â»fl?, (Ax)Â»7?V| + 1 (ap)>,J (ap) â€¢+!,J (ap)>,J(ap)Â».J+l (^).J (Â°p)Â»+l,J + (Ax)2Rvt , Kpkj + (ap)Â«,j+i (4.26) The velocity corrections are found by back-substitution. The entire procedure is summarized in the following algorithm. BRB(u*, v*, pm;ujuv,uc) Compute u' coefficient a,p(u*,v*) and residual , V(Â¿,y) Compute v' coefficient ap(u*,r*) and residual R^Â¿, V(i,j) Compute p'j, back-substitute for u\r u-+lj, v't J, vij+i V(*,j) | Â¿ + j = odd Correct all u, u, and odd p n11j â– u, j -j- cjuvv.^j (analogous corrections for Ã±Â¿+ij, Ã±t)j, Ã±Â¿j+i, and pltJ) Compute u' coefficient a,p(u,v) and residual /2â€œj, V(Â¿, j) Compute r/ coefficient avP(Ã¼,v) and residual , V(z,j) Compute pj- â– , back substitute for u'i j: uÂ¿+lj-, u-j, u'J+1 V(Â¿, j) | * + j = even Correct all u, v, and even p Ui,j T ^UV^iJ (analogous corrections for iq+ij, u,j, Uiy+i, and p,j) 97 In general the convergence rate in the multigrid context is different between SIMÂ¬ PLE and BRB. Linden et al. [50] stated a preference for the locally-coupled explicit smoother rather than pressure-correction methods. The argument the authors gave was that the local coupling of variables is better suited to produce local smoothing of residuals, i.e. faster resolution of the local variations in the solution. This is beÂ¬ lieved to allow a more accurate coarse-grid approximation of the fine-grid problem. Similar reasoning appears to have been applied in the original development [89], by Ferziger and Peric [22], and by Ghia et al. [28]. Linden et al. [50] did a simÂ¬ plified Fourier analysis of locally-coupled smoothing for the Stokes equations and confirmed good smoothing properties of the locally-coupled explicit method. Shaw and Sivaloganathan [71] have found that SIMPLE (with the SLUR solver) also has good smoothing properties for the Stokes equations, assuming that the pressure- correction equation is solved completely during each iteration. Thus there is some analytical evidence that both pressure-correction methods and the locally-coupled explicit technique are suitable as multigrid smoothers. However, the analytical work is oversimplifiedâ€”numerical comparisons are needed on a problem-by-problem basis. Sockol [80] has compared the performance of BGS. two line-updating variations on BGS, and the SIMPLE method with successive line-underrelaxation for the inner iterations. Three model flow problems were tested with different physical characÂ¬ teristics and varying grid aspect ratios: lid-driven cavity flow, channel flow, and a combined channel/cavity flow (â€œopen cavityâ€). In terms of work units, Sockol found that all four smoothers were competitive for lid-driven cavity flow over a range of Re from 100 to 5000. For the developing channel flow, BGS and its line-updating variants converged faster than SIMPLE on square grids, but as the grid aspect ratio increased SIMPLE became competitive. 98 Brandt and Yavneh [8] have developed a line-relaxation-based multigrid method which handles pressure and velocity sequentially. Good convergence rates were obÂ¬ served for â€œentering-typeâ€ flow problems in which the flow has a dominant direction and is aligned with grid lines. Line-relaxation has the effect of providing non-isotropic error smoothing properties to match the physics of the problem. Wesseling [91] anÂ¬ alyzed several line-relaxation methods, and concluded that alternating line-Jacobi relaxation had robust smoothing properties and, somewhat unexpectedly, that it was a better choice than SLUR. For pressure-based smoothers, numerical experimentation apparently has created some intuition regarding the relative performance of sequential and locally-coupled smoothers in model flow problems, but many of the issues have not been investigated systematically. Further research perhaps should not be directed toward the goal of picking one method over the other. General conclusions are unlikely because the convergence rate is dependent on the particular flow problem. Instead, both types of smoothers should continue to be implemented and tested in the multigrid context, not to determine a preference but rather to build understanding for their application to complex flow problems. The cost per iteration of BRB and SIMPLE are comparable on serial computers. If vu = uv â€” 1 and uc = 4 successive line-underrelaxation inner iterations are used, SIMPLE costs about 30% more per iteration than BGS [80]. BGS and BRB are identical in terms of run time on a serial computer. The relative cost is different on parallel computers though. Figures 4.2, 4.3 and 4.4 compare the parallel run time per iteration of BRB with SIMPLE on a 128-VU CM-5, i.e. (32 SPARC nodes each controlling 4 vector units), for a fixed number of iterations (500) of the single-grid BRB and SIMPLE solvers. The convection terms are central-differenced and, for SIMPLE, point-Jacobi inner iterations are used with 99 uu = vv = 3 and uc = 9. The problem size is given in terms of the virtual processor ratio; the largest problem size in Figures 4.2, 4.3 and 4.4 is 106 grid points. Figure 4.2 indicates that SIMPLE and BRB have virtually the same cost per 500 iterations and that this cost scales linearly with the problem size on a fixed number of processors. Figure 4.3 shows that BRB requires almost twice as much time on coefficient computations, but only about half as much on solving for the pressure changes and back-substituting. The coefficient computation cost would be exactly twice that of SIMPLE except for the small contribution from the computation of the //-equation coefficients in the SIMPLE procedure. Figure 4.4 shows the amount of time spent on computation and interprocessor communication. The interprocessor communication cost is relatively small compared to the computation cost. Also, the sum of the two is less than the total elapsed time shown in Figure 4.2, due to front-end-to-processor communication. The relative time spent overall and in computation is essentially the efficiency. Thus, the results shown in Figures 4.2-4.4 are summarized by the point-Jacobi curve in Figure 3.2. Furthermore, the breakdown into communication and computation is approximately the same for both SIMPLE and BRB, so in terms of efficiency, similar characteristics for BRB are expected as were observed in Chapter 3 for SIMPLE. In Figures 4.2-4.4 the SIMPLE timings will be different if line-Jacobi inner itÂ¬ erations are used instead of point-Jacobi inner iterations. The parallel efficiency is reduced and the actual parallel run time is greater. One line-Jacobi inner iteration (consisting of two tridiagonal solvesâ€”one treating the unknowns implicitly along horÂ¬ izontal lines and the other for the vertical lines) using the cyclic reduction method introduced in Chapter 3 takes about 8-10 times as long as one point-Jacobi iteration 100 on the CM-5. Line-Jacobi inner iterations are therefore not preferred over point- Jacobi inner iterations for use in the SIMPLE algorithm unless the benefit to the convergence rate is substantial. The line-updating variants of BRB (see [80, 87]) are even worse in comparison with BRB than the line-Jacobi SIMPLE method is in comparison with the point-Jacobi SIMPLE methodâ€”they are not suitable for SIMD computation. The line-updating variations on BGS couple pressures and velocities between control volumes along a line as well as within each control volume. By contrast, in sequential pressure-based methods, line-iterative methods are used within the context of solving the individual systems of equations, so only a single variable is involved. On the staggered grid, the unknowns which are to be updated simultaneously in the line-variant of BRB are, for a constant j line, {p2,j, u3,jiP3,ji â– â€¢ â– Â»uni-i,j,Pni-i,j}- To set up the tridiagonal system of equations for solving for these unknowns simulÂ¬ taneously requires coefficient and source-term data to be moved from arrays which have the same layout as the u and p arrays. But this data must be moved to an array(s) which has a longer dimension in the i-direction. Instead of having dimenÂ¬ sion ni, the array which contains the unknowns, diagonals, and right-hand sides has dimension 2 ni. The elements 1 :ni for the constant j line of u and the u coefficient arrays, (u, ap, a^, a]y, ag, aft/, 6â€œ), must be moved into positions l:2m:2. Similar data movement is required for the p coefficients and data. Thus, â€œSEND"-type commuÂ¬ nication will be generated during each iteration to set up the tridiagonal system of equations along the lines. This type of communication is prohibitively expensive in an algorithm where all the other operations are relatively fast and efficient. Thus, if line-relaxation smoothing is required to solve a particular flow problem for either a single-grid or a multigrid computation on the CM-5, the pressure-correction methods should be used. Otherwise, either BRB or SIMPLE-type methods can be 101 used, if time per iteration is the only consideration. With uu = uv = 3, uc = 9, and point-Jacobi inner iterations, SIMPLE and BRB have essentially the same parallel cost and efficiency. 4.3 Stability of Multigrid Iterations It is well known that central-difference discretizations of the convection terms in the Navier-Stokes equations may be unstable if cell Peclet numbers are greater than two, depending on the boundary conditions [73]. The coarse-grid level(s) have higher cell Peclet numbers. Consequently, multigrid iterations may diverge, driven by the divergence of smoothing iterations on coarse grids, if central-differencing is used. The convection terms on coarse grids may need to be upwinded for stability. However, second-order accuracy is usually desired on the finest grid. The â€œstabilization stratÂ¬ egyâ€ is the approach used to provide stability of the coarse-grid discretizations while simultaneously providing second-order accuracy for the fine-grid solution. The naive stabilization strategy is to simply discretize the convection terms with first-order upwinding on the coarse-grid levels and by second-order central-differencing on the finest grid. Unfortunately, the naive approach does not workâ€”there is a â€œmisÂ¬ matchâ€ between the solutions on neighboring levels if different convection schemes are employed, resulting in poor coarse-grid corrections. In practice divergence usually results. The coarse-grid discretization needs to be consistent with the fine-grid disÂ¬ cretization in order that an accurate approximation of the fine-grid residual equation is generally possible. In the present work a â€œdefect-correctionâ€ stabilization strategy is employed as in [80, 81, 87, 89]. The convection terms on all coarse grids are discretized by first- order upwinding. The convection terms on the finest grid are also upwinded, but a 102 source-term correction is applied which allows second-order central-difference accuÂ¬ racy to be obtained, when the multigrid iterations have converged. Another approach is to use a stable second-order accurate convection scheme, e.g. second-order upwinding, on all grid levels [74]. Shyy and Sun [74] have used different convection schemes on all grid levels and compared the convergence rates. Central-differencing, first-order upwinding, and second-order upwinding were tested for Re = 100 and Re = 1000 lid-driven cavity flow problems. Comparable converÂ¬ gence rates were obtained for all three convection schemes, whereas for single-grid computations there are relatively large differences in the convergence rates. Central- differencing was unstable for the Re = 1000 case, but a hybrid strategy with second- order upwinding on the coarsest three grid levels and central-differencing on the finer grid levels remedied the problem without deteriorating the convergence rate. Further study of this issue is conducted in Chapter 5, in which the convergence rate and staÂ¬ bility characteristics of second-order upwinding on all grid levels is contrasted with the defect-correction strategy. A third possibility is simply to add extra numerical viscosity to the physical viscosity on coarse grids. This technique has been investigated by Fourier analysis for a model linear convection-diffusion equation in [93]. The authorsâ€™ best strategy was the one in which the amount of numerical viscosity was taken to be proportional to the grid spacing on the next (finer) multigrid level. For the Navier-Stokes this brute-force approach is not expected to perform very well because the solutions on the fine grids are frequently not just a smooth continuation of the lower Reynolds number flow problems being solved on the coarse grid levels. Rather, fundamental changes in the fluid dynamics occur as Reynolds number increases. 103 4.3.1 Defect-Correction Method In the defect-correction approach, the discretized equations for a variable <\> are derived as follows. In general, the equations have the form acp4>p â€” acp(f>E + clw&w + + acg 4>s + bp (4-27) where the superscript â€œceâ€ denotes that central-differencing of the convection terms. To form the discrete defect-correction equation, the corresponding first-order up- winded discrete equation is added to and subtracted from Eq. 4.27 and rearranged to give flpVp = Â«p1 [!Â«p - Â«p'lC'p - Â« - aâ€˜i)4>E - Â« - Â«-Xv'mv - K? - Â«S)Av - (o?1 - Ã¡$) The term in brackets is equal to the difference in residuals, so Eq. 4.28 can be written a'p 4>p = a'gfc + aticf>w + a# To obtain the updated solution, the difference in residuals is lagged. Thus Eq. 4.29 for the solution at iteration counter â€œn+1â€ with the residuals evaluated at iteration counter â€œnâ€ is written auPx4>P = + g^Vjv + affc + bf + [rul - rce]n. (4.30) Moving the first five terms on the right-hand side to the left-hand side, Eq. 4.30 can be rewritten concisely as tzlln+l [râ€œir - [rcT, (4.31) 104 in which it is easily seen that satisfaction of the second-order central-difference equaÂ¬ tion discretized equations, rce â€”>â– 0, is recovered when [rul]n+1 is approximately equal to [rul]n. Table 4.1 compares the convergence rates for single-grid. SIMPLE computations using four popular convection schemes, for a lid-driven cavity flow problem. The purpose is to gain some intuition regarding the convergence properties of the defect- correction scheme. For all the cases presented in the table, the grid size was 81 x 81. The table gives the number of iterations required to converge both of the momentum equations to the level ||ru|| < â€”5.0. where the L\ norm is used, divided by the number of grid points. The inner iterative procedure for computing an approximate solution to the u, v, and p' systems of equations, during the course of the outer iterations of the SIMPLE algorithm, is listed in column 2. In the line-Jacobi method, all the horizontal lines are solved simultaneously, followed by the vertical lines, during a single inner iteraÂ¬ tion. The SLUR procedure (same technique as in Chapter 2) also alternates between horizontal and vertical lines. In addition, the grid lines are swept one at a time in the direction of increasing i or j, in the Gauss-Seidel fashion, instead of all at once as in the line-Jacobi method. The number of inner iterations for each governing equation was uu â€” uv = 3, and uc = 9 in the Re = 1000 problem. These parameters are increased to 5, 5, and 10 for the Re = 3200 flow. The inner iteration damping factor for the line-Jacobi iterative method was 0.7. For the Re = 1000 cases, the SIMPLE relaxation factors are 0.4 for the momenÂ¬ tum equations and 0.7 for the pressure. The convergence rate of defect-correction iterations is not quite as good as central-differencing or first-order upwinding, but it is slightly better than second-order upwinding. This result is anticipated for cases where 105 Flow Problem Inner Iterative Method First-order Upwinding Convect Defect Correction ion Scheme Central Differencing Second-order Upwinding Ee = 1000 Cavity Point-Jacobi 2745 3947 1769 4419 Re = 1000 Cavity Line-Jacobi 2442 3497 1543 3610 Re = 1000 Cavity SLUR 2433 3482 1534 3568 Re = 3200 Cavity Point-Jacobi 16526 > 20000 12302 > 20000 Re = 3200 Cavity Line-Jacobi 16462 > 20000 12032 > 20000 Re = 3200 Cavity SLUR 16458 > 20000 11985 > 20000 Table 4.1. Number of single-grid SIMPLE iterations to converge to ||ru|| < 10-5, for the lid-driven cavity flow on an 81 x 81 grid. The L\ norm is used, normalized by the number of grid points. central-differencing does not have stability problems, since the defect-correction disÂ¬ cretization is a less-implicit version of central-differencing. Likewise one should expect the convergence rate of SIMPLE with the defect-correction convection scheme to be slightly slower than with the first-order upwind scheme due to the presence of source terms which vary with the iterations. The method (line-Jacobi, point-Jacobi, SLUR) used for inner iterations has no influence on the convergence rate for either Reynolds number tested. From experience it appears that the lid-driven cavity flow is unusual in this regard. For most problems the inner iterative procedure makes a significant difference in the convergence rate. For the Re = 3200 cases, the relaxation factors were reduced until a converged solution was possible using central-differencing. Then these relaxation factors, 0.1 for the momentum equations and 0.3 for pressure, were used in conjunction with the other convection schemes. Actually, in the lid-driven cavity flows, the pressure plays a minor role in comparison with the balance between convection and diffusion. Consequently, the pressure relaxation factor can be varied between 0.1 and 0.5 with negligible impact on the convergence rate. The convergence rate is very sensitive to the momentum relaxation factor, however. The Re = 3200 cavity flow is hard to 106 converge, and neither the defect-correction or second-order upwind schemes succeeds for these relaxation factors. Second-order central differencing does not normally look this good either. The lid-driven cavity flow is a special case for which central- difference solutions can be obtained for relatively high Reynolds numbers due to the shear-driven nature of the flow and the relative unimportance of the pressure- gradient. For the Re = 3200 case, the convergence paths of the four convection schemes tested are shown in Figure 4.5. None of the convection schemes is diverging, but the amount of smoothing appears to be insufficient to handle the source terms in the 2nd-order upwind and defect-correction schemes for this Reynolds number. 4.3.2 Cost of Different Convection Schemes There was initially some concern that the source term evaluations in the defect- correction and/or second-order upwind convection schemes might be expensive in terms of the parallel run time. In light of Figure 4.3, it is of interest to know whether the cost per iteration is significantly increased, as this consequence might lead one to favor one convection scheme over another for considerations of run time, if both have satisfactory convergence rate characterisitics. Figure 4.6 compares the cost of computing the coefficients of the discrete u, v, and p' equations, for three convection schemes. The timings were obtained on a 32-node (128 vector unit) CM-5 for 500 SIMPLE iterations. Since the smoother and the coefficient computations are the most time-consuming tasks in the SIMPLE algorithm, the cost of the inner iterations (the â€œsolverâ€) is included for comparison purposes (the solid line). There are 15 point-Jacobi inner iterations per outer iteration, distributed 3 each on the momentum equations and 9 on the p'-system of equations. 107 The timings were obtained over a range of problem sizes, for 500 SIMPLE iteraÂ¬ tions. The x-axis in Figure 4.6 plots problem size in terms of the virtual processor ratio VP. VP is preferred over the number of grid points so that the results can be carried over to CM-5s with more processors. The coefficient cost scales linearly with problem size and, with the defect-correction scheme, requires about the same time as solving the equations. If more inner iterations were used, or the more costly line-Jacobi method was used, the fraction of the overall run time due to the computaÂ¬ tion of coefficients would decrease. The linear scaling with VP is possible due to the uniform boundary coefficient computation implementation, discussed in Chapter 3. The figure also shows that second-order upwinding of the convection terms costs more than the other schemes, by approximately 50%. Additional testing has shown that the first-order upwind, hybrid, central-difference, and defect-correction schemes all use roughly the same amount of time. More details are shown in Figure 4.7, which breaks down the time spent comÂ¬ puting coefficients into computation and interprocessor communication. Because the difference stencils are compact, only nearest-neighbor processing elements need to communicate in the calculation of the equation coefficients. These are â€œNEWSâ€- type communications on the CM-5. In the present implementation, the coefficient computations for the momentum equations require 9 NEWS communications for the defect-correction, central-differencing, and first-order upwind schemes. Second-order upwinding requires at least 13 NEWS communications. In the present implemenÂ¬ tation 17 communication operations are needed because the formulation supports nonuniform grids and therefore some geometric quantities need to be communicated in addition to the nearby velocities. The additional NEWS communication is apparÂ¬ ent in Figure 4.7. Similarly, the second-order upwind scheme involves more compuÂ¬ tation than the other schemes. 108 Coincidentally, the additional computation and interprocessor communication of the second-order upwind convection scheme offset each other in terms of their affect on the parallel efficiency. With either convection scheme the trend is essentially the same, Figure 4.8. Figure 3.2 gave the variation of E with VP for central-differencing. 4.4 Restriction and Prolongation Procedures The discretization of the convection terms on coarse grids is a key issue because the coarse grid problem must be a reasonable approximation to the fine-grid discretized equation, in order to obtain good corrections. In addition, for the formulation given in the background section, one must also say how the coarse-grid source terms are computed, and how the corrections are interpolated to the fine grid. The restricÂ¬ tion and prolongation procedures affect both the stability and convergence rate. In this section, three restriction procedures and two prolongation procedures have been compared on two model problems with different physical characteristics to assess the effect of the intergrid transfer procedures on the multigrid convergence rate. For finite-volume discretizations, conservation is the natural restriction procedure for the equation residuals, because the terms in the discrete equations represent integrals over an area. The method of integration for source terms determines the actual restriction procedure. For piecewise constant treatment of source terms in a cell-centered finite-volume discretization, the mass residual in a coarse-grid control volume is the sum of the mass residuals in the four fine-grid control volumes which comprise the coarse-grid control volume. This restriction procedure is used for the residuals of the continuity equation in every case tested. If the mass residual is summed, and and v% are restricted by cell-face averaging (described below), the right-hand side of Eq. 4.22 is identically zero [80], which implies that the velocity field on coarse grids also satisfies the continuity equation, in addition 109 to the velocity field on the finest grid. However, it is not necessary to have identically zero coarse-grid source terms, even in the continuity equation. Restriction procedure â€œ3â€ obtains the initial coarse-grid solutions not by restrictÂ¬ ing the solutions, but instead by taking the most recently computed values on the coarse grid. These values will be from the previous multigrid cycle. The Â«-momentum equation residuals are summed over the six fine-grid u control volumes which comprise the coarse-grid u control volume under consideration. Only half the contribution is taken from the cell-face neighbor u control volumes due to the staggered grid. For the restriction procedure denoted â€œ1,â€ u, v, and the momentum equation residuals are restricted by cell-face averaging. Cell-face averaging refers to the averÂ¬ aging of the two fine-grid u velocity components immediately above and below the coarse-grid u location, which are on the same coarse-grid p control volume face. SimÂ¬ ilar treatment is applied to v. The coarse-grid pressures are obtained by averaging the four nearest fine-grid pressures. The restriction procedure â€œ2â€ indicates a weighted average of six fine-grid u veÂ¬ locity components, the cell-face ones and their nearest-neighbors on either side. The cell-face fine-grid u velocity components contribute twice as much as their neighbors. Similar treatment is applied for v, and for the momentum equation residuals. The coarse-grid pressures are obtained by averaging the four nearest fine-grid pressures, as in restriction procedure 1. For the prolongation procedures, â€œ1â€ and â€œ2â€ indicate bilinear and biquadratic interpolation, respectively. The bilinear interpolation procedure is identical to that used by Shyy and Sun [74], in which the two nearest coarse-grid corrections along a line x â€” constant (for u) are used to compute the correction at the location of the fine-grid u velocity component, by linear interpolation. Similar treatment is adopted for v corrections. To compute the corrections on the â€œin-betweenâ€ fine-grid lines the no available fine-grid corrections are interpolated linearly. Corrections for pressure are interpolated linearly from the four nearest coarse-grid values. The biquadratic interpolation procedure, â€œ2,â€ is similar to the procedure used by Bruneau and Jouron [12]. It finishes in exactly the same way as the bilinear inÂ¬ terpolation, but is preceded by a quadratic (instead of linear) interpolation in the y-direction, and an averaging in the ^-direction. Thus, the three nearest correction quantities on the coarse grid (above and below the fine-grid u location) are used to interpolate in the y-direction for a correction located at the position of the fine-grid u velocity component. After this y-direction interpolation there are two corrections defined on each face of the coarse-grid u control volumes, at the locations correspondÂ¬ ing to the locations of the fine-grid u velocity components. These are injected to give the fine-grid corrections at these points after a weighted averaging in the x-direction. For example, on a uniform grid this pre-injection averaging goes like: ^c,corr (11 J) â€” i^c.corr (I T 11 J) T 2 UCCorr (11 J) T ^c.corr (I l)*^))/4, (4.32) where uCtCorr and the capitalized indices indicate that the correction quantities are still defined on the coarse gridâ€”they are positioned to correspond with the fine-grid u locations. After the averaged corrections are injected to the fine grid, the fine- grid corrections are defined along every other line x = constant. The corrections on â€œin-betweenâ€ lines are linearly interpolated from the injected, averaged correcÂ¬ tions. Similar treatment is adopted for the v corrections. Corrections for pressure are interpolated biquadratically from the nine nearest coarse-grid values. Table 4.2 below compares the various intergrid transfer procedures in terms of the work units required to reach a prescribed convergence tolerance on the finest grid level. The notation (p,r) indicates the number of the prolongation and restriction procedures adopted. The convergence tolerance on the fine grid is prescribed by Ill an estimate of the truncation error of the fine-grid discretization, which is derived in Chapter 5. The criterion is typically not very stringent, so the table results best reflect differences in the initial convergence rate instead of the asymptotic convergence rate. Number of work units to converge (P'r) Re = 1000 Cavity Re = 400 Back-Step V (2,1) V(3,2) V (2,1) V(3,2) (1,1) 19.0 23.6 123.2 95.7 (2,1) 21.8 28.5 110.0 166.6 (1,2) 16.9 24.4 168.9 181.7 (2.2) 20.2 20.5 263.5 122.4 (1,3) 12.7 13.6 div 51.8 (2,3) 14.1 13.8 239.5 59.6 Table 4.2. The effect of different restriction and prolongation procedures on the conÂ¬ vergence rate of the pressure-correction multigrid algorithm, for a 7-level cavity flow problem with a 322 x 322 fine grid, and for a 5-level symmetric backward-facing step flow with a 322 x 82 fine grid. The defect-correction approach is used. Numerical experiments with the number of pre- and post-smoothing iterations have shown that for the cavity flow, V(2,l) cycles provide enough smoothing. V(3,2) cycles are needed for the symmetric backward-facing step flow computation. With less smoothing the number of work units to reach convergence generally increases even though the number of work units per cycle is smaller. The restriction procedure used appears to be very important to the convergence rate in either flow problem. The restriction procedure 3 appears to perform better than 1 or 2. The discussion presented earlier suggested this result. However, since the residuals are summed instead of averaged they are typically larger, with more spatial variation also. As a result, more smoothing iterations are needed to ensure stability of the multigrid iterations. For r = 3, it appears that the bilinear interpolation procedure (p = 1) converges slightly faster than the biquadratic procedure. 112 The performance of the other restriction procedures appears to depend on the prolongation procedure. In both problems the best results for r = 1 or r = 2 are obtained when the corresponding (p = 1 or p = 2) prolongation procedure is used. In the backward-facing step flow, the results for cell-face averaging (r = 1) are better than the six-point averaging by a significant amount. The same is true for the cavity flow but to a lesser degree. The effect of Reynolds number for each flow problem should be considered in future work. Figures 4.9, and 4.10 give a different look at the relative performance of the 1 and 3 restriction procedures, cell-face averaging of solutions and residuals contrasted with summation of residuals only. The focus is on the asymptotic convergence rate as opposed to the initial convergence rate considered in Table 4.2. The u-momentum equation average residual (the L\ norm divided by the number of grid points) is plotted on each grid level against work units. V(3,2) cycles and bilinear interpolation (p = 1) were used for the symmetric backward-facing step flow calculation. The computations have been carried far beyond the point at which convergence was declared in Table 4.2. The dashed line shows the estimated truncation error on the fine grid used to declare convergence for the table. Brandt and Yavneh have argued that this level of convergence should be sufficient [9]. Further multigrid cycles reduce the algebraic error but not necessarily the differential error. With restriction procedure 1, Figure 4.9, the initial multigrid convergence rate is rapid, but levels off significantly after about 100 work units. This apparently slow asymptotic multigrid convergence rate is still much better than the single-grid converÂ¬ gence rate for this flow problem, indicating that there is some benefit being obtained from the coarse-grid corrections with the restriction procedure 1. The corrections are evidently not as large as with restriction procedure 3 (Figure 4.10), because this case shows no reduction in the initial rapid convergence rate. It has been verified that 113 the convergence rate is maintained until the level of double-precision roundoff error (-15.0) is reached, although the convergence path is shown only down to -8.0. These figures support the earlier observation that the restriction procedure 3 is appropriate to the finite-volume discretization. The difference between the performance of the restriction procedures 1 and 3 is even more dramatic in the lid-driven cavity flow, Figures 4.11 and 4.12. The convergence rate of the present multigrid method appears to be comparable to other results in the literature. Sockol [80] found that roughly 30 work units were needed to obtain convergence for the lid-driven cavity flow at Re = 1000, for both BGS and SIMPLE. The residuals were summed as in restriction procedure 3, but the variables were also restricted, by cell-face averaging. W(l,l) cycles were used. Shyy and Sun [74] needed many more work units to reach convergence, using V cycles at the same Reynolds number but with less resolution on the fine grid (81 x 81). The restriction procedure 1 was used. The convergence criterion was tighter, and there were procedural differences from the present work and that of Sockol which may also account for the differences. 4.5 Concluding Remarks Multigrid techniques are potentially scalable parallel computational methods, both in the numerical sense and the computational sense. The key issue for applying multigrid techniques to the incompressible Navier-Stokes equations is the connection between the evolving solutions on the various grid levels, which includes the transfer of information between coarse and fine grids, i.e. the restriction and prolongation procedures, and the formulation of the coarse-grid problem, i.e. the choice of the coarse-grid convection scheme. These factors also influence the stability of multigrid iterations. 114 The restriction procedure for finite-volume discretizations should be summing of residuals. Also, it was found unnecessary to restrict the solution variables. The convergence rate in both types of flow problems, shear and pressure-driven, were significantly accelerated when the residuals were summed instead of averaged. HowÂ¬ ever, because the residuals are larger, more smoothing is found to be necessary to avoid stability problems, in the symmetric backward-facing step flow. The bilinear prolongation procedure appears to be preferrable to the biquadratic prolongation procedure. The convergence rates which have been achieved in the model problems are comparable to other results in the literature. In terms of cost per iteration, it appears that the pressure-correction type smoother is comparable to the locally-coupled explicit method on the CM-5, whereas for serial computations the latter has been favored by some [80]. Both algorithms consist of basically the same operations, with roughly twice as much influence on the parallel run time from the coefficient computations, for BRB. The coefficient computation cost is comparable to the smoothing cost for the SIMPLE method, but for BRB the former is the dominant consideration. In that respect, the uniform implementation for boundary coefficient computations described in Chapter 3 and the choice of conÂ¬ vection scheme are very important considerations. Using the second-order upwind scheme, the cost per iteration of SIMPLE, assuming 3, 3, and 9 point-Jacobi inÂ¬ ner iterations, is roughly twice as much compared to the defect-correction scheme, although there is negligible effect on the parallel efficiency. 115 Level 4 (fine grid) Level 3 Level 2 Level 1 (coarse grid) (3) = 3 smoothing iterations Figure 4.1. Schematic of a V(3,2) multigrid cycle, which has three smoothing iteraÂ¬ tions on the â€œdownstrokeâ€ of the V and 2 smoothing iterations on the â€œupstroke.â€ 116 Smoother Comparison Figure 4.2. Comparison of the total parallel run time for SIMPLE and BRB on a 128 vector-unit CM-5 for 500 iterations over a range of problem sizes. The flow problem which was timed was Re = 1000 lid-driven cavity flow. 117 Smoother Comparison VP Figure 4.3. Comparison of the parallel run times for SIMPLE and BRB, decomposed into contributions from the coefficient computations and the solution steps in these algorithms. The time are obtained on a 128 vector-unit CM-5 for 500 iterations over a range of problem sizes. The convection terms are central-differenced. 118 Smoother Comparison 400 300 | 200 100 Ãœ X o SIMPLE o x Box Red-Black X - o Node cpu - X - b o Comm (NEWS) O O ox X 8 X L 0 5000 VP 10000 Figure 4.4. Comparison of the parallel run time for SIMPLE and BRB, decomÂ¬ posed into contributions from parallel computation and nearest-neighbor interprocesÂ¬ sor communication (â€œNEWSâ€). The timings were made on a 128 vector-unit CM-5 for 500 iterations over a range of problem sizes. 119 Single-Grid Convergence Paths for Re=3200 Case Number of Iterations Figure 4.5. Decrease in the norm of the ii-momentum equation residual as a function of the number of SIMPLE iterations, for different convection schemes. The results are for a single-grid simulation of Re = 3200 lid-driven cavity flow on an 81 x 81 grid. The alternating line-Jacobi method is used for the inner iterations. The results do not change significantly with the point-Jacobi or the SLUR solver. 120 Cost of Coefficient Computations Figure 4.6. Comparison between two convection schemes, in terms of parallel run time. The total (computation + communication) time spent computing coefficients over 500 SIMPLE iterations, on a 128-VU CM-5, is plotted against the virtual proÂ¬ cessor ratio, VP. â€œSolver timeâ€ is the time spent on 15 point-Jacobi inner iterations per SIMPLE iteration, 3, 3, and 9 for the u, u, and p' systems of equations. It is just coincidental that, for the defect-correction and central-difference cases, the coefficient computations and the solver time are about equal. 121 NEWS & CPU Costs in Coefficient Computations 450 400 350 ^ 300 c o 250 a> w ^200 (D Â¡1 150 100 50 0 0 x 2nd-order upwind CPU + 2nd-order upwind NEWS o Defect-correction CPU * Defect-correction NEWS x o X A. + JL X + 2000 4000 6000 VP x + X 8000 10000 Figure 4.7. For the second-order upwind and defect-correction schemes, the time spent in coefficient computations for 500 SIMPLE iterations is decomposed into conÂ¬ tributions from computation, denoted â€œCPUâ€, and from nearest-neighbor interproÂ¬ cessor communication, denoted â€œNEWSâ€. These quantities are plotted against the virtual processor ratio, VP. Times are for a 128-VU CM-5. 122 CM-5 SIMPLE Code: E vs. VP for 128 VUs 1 0.8 LU >. o 0) 0.6 o Hâ€” LU J? 0.4 CO 03 Q_ 0.2 0 0 2000 4000 6000 8000 10000 VP Figure 4.8. Parallel efficiency, E for a range of problem sizes. E = Ti/npTp, where T\ is the serial execution time, estimated by multiplying the measured computaÂ¬ tion time per processor by the number of processors, np. Tp is the elapsed CM-5 run time, including computation, interprocessor and front-end-to-processor types of communication. - 1 X 1 o X 1 1 o 9 X a o - x 2nd-order upwind scheme < D o Defect-correction scheme Solver= point-Jacobi 1 1 iterations (3, 3, and 9) 1 1 123 Level 5 Level 4 Level 3 Level 2 Level 1 Work Units Residual norm Truncation error norm Re = 400 Symmetric Back-Step FMG-FAS V(3,2) cycles 321 x 81 grid, 5 levels Defect-correction scheme Point-Jacobi solver (3,3,9) Relax, factors (.7,.7,.5) (P,r) =(1,D Figure 4.9. Convergence path on each grid level for a 5-level V(3,2) multigrid cycle. The fine grid is 322 x 82. The flow problem is a Re = 400 symmetric backward-facing flow. Bilinear interpolation (p = 1) and cell-face averaging for restriction (r = 1) are used. 124 Level 5 Level 4 Level 3 Level 2 Level 1 Work Units Residual norm â€” Truncation error norm Re = 400 Symmetric Back-Step FMG-FAS V(3,2) cycles 321 x 81 grid, 5 levels Defect-correction scheme Point-Jacobi solver (3,3,9) Relax, factors (.7,.7,.5) (P,r) =d,3) Figure 4.10. Convergence path on each grid level for a 5-level V(3,2) multigrid cycle. The fine grid is 322 x 82. The flow problem is a Re = 400 symmetric backward-facing flow. Bilinear interpolation (p = 1) and summation of residuals for restriction (r = 1) are used. Residual norm Residual norm Residual norm 125 Level 7 Level 5 Level 3 Work Units Level 6 Level 4 Residual norm Truncation error norm Re = 1000 Lid-Driven cavity FMG-FAS V(2,l) cycles 321 x 321 grid, 7 levels Defect correction scheme Point-Jacobi solver (3,3,9) Relax, factors (.7,.7,.5) (p,r) = (1.1) Figure 4.11. Convergence path on each grid level for a 7-level V(2,l) multigrid cycle. The fine grid is 322 x 322. The flow problem is Re = 1000 lid-driven cavity flow. Bilinear interpolation (p = 1) and cell-face averaging for restriction (r = 1) are used. 126 Level 7 Level 6 Level 5 Level 4 Level 3 Work Units Residual norm Truncation error norm Re = 1000 Lid-Driven cavity FMG-FAS V(2,l) cycles 321 x 321 grid, 7 levels Defect correction scheme Point-Jacobi solver (3,3,9) Relax, factors (.7,.7,.5) (P,r) = (1,3) Figure 4.12. Convergence path on each grid level for a 7-level V(2,l) multigrid cycle. The fine grid is 322 x 322. The flow problem is Re = 1000 lid-driven cavity flow. Bilinear interpolation (p = 1) and summation of residuals for restriction (r = 3) are used. CHAPTER 5 IMPLEMENTATION AND PERFORMANCE ON THE CM-5 This chapter describes the implementation on the CM-5 of the multigrid method studied previously, and applies the parallel code to two model flow problems to assess the performance both in terms of the convergence rate and the cost per iteration. The major implementational consideration for the CM-5 multigrid algorithm is the storage problem. The starting procedure by which an initial guess is generated for the fine grid is an important practical technique whose cost on parallel computers is of interest. Also, the starting procedure is important in the sense that the initial guess can affect the stability of the subsequent multigrid iterations and the convergence rate. The cycling strategy is discussed next. It also affects both the run time and the convergence rate. Because of the nonneglible smoothing cost of coarse grids, the comparison between V and W cycles in terms of the time per cycle is different than on serial computers and needs to be assessed for the CM-5. The purpose of the chapter is to provide some practical guidance regarding the use of the numerical method on the CM-5, now that the choice for the smoother, the coarse-grid discretization, and the restriction and prolongation procedures has been addressed. Finally, the computational scalability of the parallel implementation is studied using timings for a range of problem sizes and numbers of processors. With the experience gained with regards to the choice of algorithm components and practiÂ¬ cal techniques, this information gives a clear picture of the potential of the present approach for scaled-speedup performance on massively-parallel SIMD machines. 127 128 5.1 Storage Problem Multigrid algorithms pose implementational problems in Fortran, because the language does not support recursion. A variable number of multigrid levels must be accommodated but care must be taken not to waste memory. Let NI(k) and NJ(k) be arrays denoting the grid extents on the fcth multigrid level, where k â€” 1 refers to the coarsest grid and k = kmax is the finest grid. The dimension extents on the fine grid are parameters of the problem. For an array A, the different grid levels are made explicit by adding a third array dimension. This is a natural albeit naive storage declaration, PARAMETER (NI(kmax) = 1024, NJ(kmax) = 1024, kmaT = 7) REAL*8 A( NI(kmax), NJ(kmax), kmax ) Unfortunately, this approach wastes storage because every grid level is dimenÂ¬ sioned to the extents of the finest grid. The coarse grids are significantly smaller, though, decreasing in size by factor of 4 for each level beneath the top level (the fine grid). The total amount of memory used in this approach is the number of arrays, narray, multiplied by the storage cost of each array, StovciQc NI (kmax) N J (kmax)kmaxnarray (5.1) The actual storage needed is only ATTâ€ž.\ AT Tt 1-\- VI(kmax)NJ(kmax) Storage / v A I(k) A J(k)Tlarray / v 9(A:max â€” k) ^array (^*2) k= 1 k= 1 Â¿ TnaX The actual storage needed approaches (4/3)NI(kmax)NJ{kmax)narray as kmax inÂ¬ creases. Thus the wasted storage is (kmax â€” 4/3)NI(kmax)NJ(krnax)narray when the naive approach is used. Clearly this can become the dominating factor very quickly as the number of levels increases. 129 One efficient solution for serial computation is to declare a 1-d array of sufficient size to hold all the data on all levels and to reshape it across subroutine boundaries, taking advantage of the fact that Fortran passes arrays by reference. This practice is typical in serial multigrid algorithms [63]. A 1-d array section of the appropriate length for the grid level under consideration is passed to a subroutine where it is received as a 2-d array with the dimension extents NI(k) x NJ(k). On serial computers, this reshaping of arrays across subroutine boundaries is posÂ¬ sible because the physical layout of the array is linear in the computerâ€™s memory. On distributed memory parallel computers like the CM-5, however, the storage problem is not so easily resolved because the data arrays are not physically in a single proÂ¬ cessor memory, they are distributed among the processors. Instead of being passed by reference as is the case with Fortran on serial computers, data-parallel arrays are passed to subroutines by â€œdescriptorâ€ on the CM-5. The array descriptor is a front- end array containing 18 elements. The descriptor contains information about the array being described: the layout of the physical processor mesh, the virtual subgrid dimensions, the rank and type of the array, the name and so on. On the CM-5 the storage problem is resolved using array â€œaliases.â€ Array aliasing is a form of the Fortran EQUIVALENCE function used on serial computers. In the multigrid algorithm, storage for each variable is initially declared for all grid levels, explicitly referencing the physical layout of the processors. For example, an array A with fine-grid dimension extents NI(kmax) x NJ(kmax), is declared as follows for a 128-VU CM-5 with the processors arranged in an (nlp â€” 8) x (nJp = 16) mesh: PARAMETER (Nserial = (4/3)NI(kmax)NJ(kmax)/np, n'p = 8, nj = 16) REAIA8 A( N,eriai,n'p,nj, ) Actually, the factor 4/3 needs to be increased slightly to account for â€œarray padding.â€ Each physical processor must be assigned exactly the same number of 130 virtual processors in the SIMD model, since all processors do the same thing at the same time. Thus, in general the array dimensions on each level must be â€œpaddedâ€ to fit exactly onto the processor mesh. For example, an 80 x 80 fine grid with 5 multigrid levels has coarse grids with dimensions 40 x 40, 20 x 20, 10 x 10 and 5x5. To fit onto the processor mesh with exactly the same subgrid shape and size for each physical processor, assuming an 8 x 16 processor mesh, the storage allocated must be 88 x 96 + 48 x 48 + 24 x 32 + 16 x 16 4- 8 x 16 (on the coarsest grid VP = 1). Thus the actual declared storage needs to be slightly more than that shown above. The array A is mapped to the processors using the compiler directives discussed in Chapter 3. The first dimension extent of A is the actual storage needed per physical processor. It is laid out linearly in each physical processorâ€™s memory by the :SERIAL specification in the LAYOUT compiler directive (recall Chapter 3 example). The latter two dimensions are parallel (:NEWS), laid out across the physical processor mesh. Then, to access the A arrays corresponding to each grid level, array aliases (alterÂ¬ nate front-end array descriptors for the same physical data) are created as described in Chapter 3. For example, an equivalence is established between the â€œarray sectionâ€ A(l:88*96/(8*16), 1:8,1:16) and another array with dimensions (1:88,1:96). In this way arrays can be referenced inside subroutines as if they had the dimensions of the alias, with both dimensions parallel. In this case a (:NEWS,:NEWS) layout of A(88,96) can be declared, even though in the calling routine the data come from an array of a different shape. This feature, array aliasing, is relatively new in the CM-Fortran compiler evolution (version 2.1-Beta [84]) and has not yet been implemented by MasPar in their compiler. Previous multigrid algorithms on SIMD computers were restricted to either the naive approach or explicit declaration of arrays on each level [18]. The latter approach is 131 extremely tedious and leads to very large front-end executable codes, making front- end storage a concern. Thus, the present technique for getting around the multigrid storage problem, although requiring some programming diligence, is critical because it permits much larger multigrid computations to be attempted on SIMD-type parallel computers. As observed in Chapter 3, for the CM-5, problem sizes of the order of the largest possible problem sizes are necessary to obtain good parallel efficiencies. 5.2 Multigrid Convergence Rate and Stability The â€œfull multigridâ€ (FMG) startup procedure [11] is shown in Figure 5.1. It begins with an initial guess on the coarsest grid. Smoothing iterations using the pressure-correction method are done until a converged solution has been obtained. Then this coarsest-grid solution is prolongated to the next grid level and multigrid cycles are initiated (at level 2, the â€œnext-to-coarsestâ€ grid level). Cycling at this level continues until some convergence criterion is met. The solution is prolongated to the next finer grid and multigrid cycling resumes. This process is repeated until the finest grid level is reached. The converged solution on level kmax â€” 1, after interpolation to the fine grid, is a much better initial guess than is possible otherwise. The alternative is to use an arbitrary initial guess on the fine grid. For Poisson equations, one V cycle on the finest grid is frequently sufficient to reach a converged solution, if the initial guess is obtained by the FMG procedure. The benefit to the convergence rate of a good initial guess more than offsets the cost of the V cycles on coarse grids leading up to the finest grid level. For Navier-Stokes equations the cost/convergence rate tradeoff still favors using the FMG procedure, on serial computers. For parallel computers, however, the cost of the FMG procedure is more of a concern, due to the inefficiencies of smoothing the coarse grids, and the potential need for many coarse-grid cycles. 132 On SIMD computers, the smoothing iterations on the coarse grid levels have a hxed baseline time set by the communication overhead of the front-end-to-processor type. Thus, the cost of the FMG procedure is increased compared to serial comÂ¬ putation because coarse-grid smoothing is relatively more costly (less efficient) than fine-grid smoothing. It becomes important, with regards to cost, to minimize the number of coarse-grid cycles, without sacrificing the benefit of a good initial guess to the multigrid convergence rate. Tuminaro and Womble [88] have recently modelled the parallel run time of the FMG cycle on a distributed memory MIMD computer, a 1024-node nCUBE2. They developed a grid-switching criterion to account for the inefficiencies of smoothing on coarse grids. The grid-switching criterion effectively reduces the number of coarse- grid cycles taken during the FMG procedure. They have not yet reported numerical tests of their model, but the theoretical results indicate that the cost/convergence rate tradeoff can still favor FMG cycles for multigrid methods on parallel computers, with their technique. In the next section a truncation error estimate is developed and then used to control the amount of coarse-grid cycling in the FMG procedure. The validity and the numerical characteristics of the truncation error estimate are addressed. In addition to the cost of obtaining the initial guess on the fine grid, the quality of the initial guess can affect both the convergence rate and the stability of the subsequent multigrid iterations, depending on the flow problem and the coarse-grid convection scheme, i.e. the stabilization strategy. The performance of the truncation error criterion in this regard is also studied. 133 5.2.1 Truncation Error Convergence Criterion for Coarse Grids The goal of a given discretization and numerical method is to obtain an approxÂ¬ imate solution to Eq. 4.6, vh, which nearly satisfies the differential equation, i.e. to achieve \\Ahu-Ahvh\\ factors due to the grid distribution, resolution, the discretization of the nonlinear terms and the proper modelling and specification of boundary conditions. Thus the conservative philosophy is usually adoptedâ€”assume that the discretized equation is a good approximation to the differential equation and seek the exact solution to the discrete equation, i.e. seek algebraic convergence, ||Ahuh - Ahvh|| = ||Sh - Ahvh|| = ||r/l|| < e, (5.4) again choosing the level t to accommodate any imposed constraints on the run time. Eq. 5.4 is applied on the finest grid in a multigrid computation, the level on which the solution is desired. The coarse grid solution obtained in the FMG procedure has only one purposeâ€” to yield a good initial guess on the fine grid. The â€œbestâ€ initial guess is the one that allows Eq. 5.4 to be satisfied on the fine grid quickest. The corresponding coarse-grid solution from which the fine-grid initial guess is obtained may or may not satisfy Eq. 5.4 with t = 0 itself. It is not always beneficial to the fine-grid convergence rate to obtain the coarse-grid solution to strict tolerances. The utility of a coarse-grid solution for the purpose of providing a good initial guess on the fine grid depends more on the difference in the truncation errors of the Q2h and Qh approximations than it does on the accuracy of the coarse-grid soluÂ¬ tion. For example, in highly nonlinear equations or in problems where grid levels are 134 coarsened by factors greater than two, it is immediately apparent that the solution accuracy in the coarse-grid solution to the discrete problem cannot translate into a truly accurate initial guess on the fine grid no matter how accurately the coarse grid problem is solved. The usefulness of the coarse-grid solution depends on the smoothness of the physical solution and the prolongation procedure. Consequently, one expects that the most cost-effective procedure for controlling the FMG cycling will be obtained with a particular set of coarse-grid tolerances that depend on the flow characteristics. Thus the goal should be to discontinue the FMG cycles on a particular coarse grid level when Eq. 5.3 is satisfied. Frequently Eq. 5.3 is satisfied before Eq. 5.4. Similar arguments have been made by Brandt and Taâ€™asan [7]. Using the definitions of the truncation error, Eq. 4.2, and the residual, the triangle inequality gives ||A2hu - A2hv2h|| < \\A2hu - A2hu2h|| + ||A2hu2h - A2hv2h\\ = ||r2/l|| + ||r2/l||. (5.5) Thus, if r2Ã = e/2, (5.6) Eq. 5.3 can be satisfied if the residual is less than the truncation error, I Jih\\ ^ 11 _2/i 11 r <. r II. (5.7) Eq. 5.7 is the criterion applied to the coarse grids, while Eq. 5.4 is retained for the finest level. To develop an estimate for ||r2/i|| in Eq. 5.7, consider an example case of a 1-D nonlinear convection diffusion equation with a constant or position-dependent source term, du d2u UYx~V~d^ = 5' (5.8) 135 For a finite-difference discretization with central-differencing for both derivative terms, the truncation error at grid point â€œiâ€ on the grid with spacing h is given by 'Â«,+1 - 2ut + it,_i' ^Â¿+1 1 2h h2 -5,- = r* uh2 (5.9) /i2 -li; 17,17, + ... 24 * 6 â€™ where u is the differential solution at the position x â€” ih. Similarly on the grid with spacing 2h, fui+i ~ Â«7-1 UI { 2h fui+i â€” 2uÂ¡ + u/_i \ "l v >-5' = T (5.10) 4 uh2 â– 4h2 Â»/ = ^ru' - xu,â€œ'+ â– - The grid points x = Ih and x = ih correspond, but 1+1 refers to the point at x = xÂ¡ T 2h, whereas i+1 refers to the point at x = x, -f h. Assuming the high-order terms are negligible (debatable for fluid flow problems unless the solution is very smooth), and subtracting the first equation from the second (at the grid points of n2h), one obtains /Â«/+1 - 2Ul + 17/â€”1 Â«7+1 â€” Â«7-1 UI{ 2h )~v\ h2 ( Â«t-M - Â«Â¿-1 N (Â«Â¿+1 - 2li, -I- Ui-i Ul V 2h ) U V h2 -SÂ¡ s, (5.11) 3r^ In operator notation, A2hu - Ahu - Sh] = 3rfc. (5.12) Substituting the most current approximation vh for u (at the coarse-grid grid points), and the approximate values v2h = /2/lu/l, this expression becomes A2h(I2hvh) - S' Ih Ahvh -Sn I ~3rft. (5.13) 136 The term in brackets is just the residual rh (at the corresponding coarse-grid grid point). For finite-difference discretizations this residual is presumed to be accurately approximated by I2hrh. Thus the truncation error of the fine-grid discretization, estimated at the coarse-grid grid points, is A2h(I2hhvh) - S2h - I2hhrh Th ~ (5.14) This expression, however, is merely the numerically derived part of the coarse-grid source term, S2nhumerical in Eq. 4.14. Thus S2h h ^ numerical (5.15) The convergence criteria based on this truncation error estimate, Eq. 5.7, becomes < 02 h ^numerical (5.16) The norms used on each side of the equation should be divided by the appropriate number of grid points (since they are defined on different grid levels), so that the quantities represented are comparable. The L\ norm is used hereâ€”on a grid with N2 points, the L\ norm of a vector v is IMI = 5Z all i,j . N2 (5.17) Eq. 5.16 is very convenient. It is a way of setting the coarse-grid tolerances in the FMG procedure automatically. Also, since the additional coarse-grid term S2^mertcal is already computed as part of the coefficient computations precediing the coarse-grid smoothing, there are no new quantities to be computed and monitored. 5.2.2 Numerical Characteristics of the FMG Procedure The following issues are addressed: the validity/utility of the analysis above leadÂ¬ ing to Eq. 5.16, the performance of the resulting FMG procedure based on the trunÂ¬ cation error convergence criterion in terms of the cost and the initial residual level 137 on the fine grid, and the characteristics of the convergence path through the FMG cycling as a function of the flow problem and the coarse-grid convection scheme. Two flow problems with very different physical characteristics are considered, the lid-driven cavity flow at Reynolds number 5000 and a symmetric backward-facing step flow at Reynolds number 300. Streamlines, velocity, vorticity and pressure contours for the two model flow problems are shown in Figures 5.2 and 5.3, to clarify the problem specification and bring out their different physical features. In the streamline plots, the contours both inside and outside the recirculation regions are spaced evenly. However, because the recirculation regions are fairly weak in both problems, the spacing between contour levels is set to be smaller within the recirculation regions in order to bring out the flow pattern. The lid-driven cavity flow is a recirculating flow where convection and crossÂ¬ stream diffusion balance each other in most of the domain and the pressure gradient is important only in the upper-left corner. In contrast, the symmetric backward-facing step flow is aligned with the grid for much of the domain. The pressure gradient balances viscous diffusion as in channel-type flows. These problems are challenging in different ways and are representative of much broader cross-sections of interesting flow situations. Figures 5.4-5.7 show the convergence path of the Â«-momentum residual in the lid-driven cavity flow for different coarse-grid convergence criteria. The residual is plotted for the current outermost level, during the FMG procedure. Also the plot is continued for the first three multigrid cycles on the finest grid level to show the initial multigrid convergence rate on the fine grid. The finest grid level was 321 x 321 and seven multigrid levels were usedâ€”the coarsest grid is 6 x 6. The defect-correction approach was used, first-order upwinding on coarse grids and defect-correction on the finest level. V(3,2) cycles with bilinear interpolation for the prolongation procedure 138 and restriction procedure 3, piecewise-constant summation of the residuals only, were used. The relaxation factors were uuv = 0.5 and u>p = 0.5, and point-Jacobi inner iterations were used, with vu = vv = 3, and uc = 9. In the symmetric backward-facing step results given below, the same procedures are used, except in the smoother the relaxation factors are luuv = 0.6 and lov = 0.4. The fine grid is 321 x 81 and five multigrid levels are used. In Figure 5.4, the truncation error criterion Eq. 5.16 is applied, with the denomÂ¬ inator set to 1. This is the â€œrightâ€ denominator according to the analysis behind Eq. 5.16, since the outermost levels during the FMG cycling on coarse grids are first- order accurate in the convection term, provided convection is important in the flow problem. The tolerances given by the truncation error criterion are graded, because the truncation error is larger on coarser grids. The spacing between the levels is uneven, though, and depends on the evolving solution. For the cavity flow, ||rfc|| in Eq. 5.16, with the denominator equal to 1, converged to +0.2, -0.4, -1.2, -1.9 and -2.6 for levels 2 through 6. On the finest grid the truncation error estimate converges to -3.0. The figure shows a jump in the residual level going from coarse grid to fine grid of approximately -0.6 between any two successive levels. This jump is just logiol/4. Physically the equation residuals represent integrated quantities in the finite-volume discretization. Thus, whether on the coarse or the fine grid, the net residual (L\ norm) should be roughly the same (or greater, because the bilinear/biquadratic inÂ¬ terpolations considered here should not be expected to improve the solution since they are not derived from the physics). In the norm used here the sum of the residuals is divided by the number of grid points. Thus, in the best case one would anticiÂ¬ pate the result which has been obtained, with the factor of 4 decrease in the average residualâ€”the fine-grid control volumes are a factor of 4 smaller than the coarse-grid 139 control volumes. The fact that the maximum jump is achieved indicates that the order of the prolongation procedure is sufficient for the flow problem. In Figure 5.8 the corresponding case for the symmetric backward-facing step flow is shown. The jump in the average u residual between levels is about -0.4. Similar observations hold for second-order upwinding in both flow problems, using the truncation error criterion with the denominator set equal to three. Thus, the results obtained are plausible and one would expect, about the best results which are possible. Figure 5.5 shows the effect of applying a more stringent coarse-grid convergence criterion. In this case the truncation error estimate is again used but with the deÂ¬ nominator set to five. A slight improvement in the initial level of the residual on the finest grid is obtained. After 1 fine-grid cycle, the residual is -3.5 compared to -3.25 for the 1-FMG cycle. However, tightening the coarse-grid tolerances even further does not give any benefit. For example, Figure 5.6 shows the FMG convergence path when the coarse-grid residual is driven down to a specified value on each level, i.e. when dr'll < t (5.18) is applied, with t = â€”3.0 in Figure 5.6. Also, in the subsequent figure, Figure 5.7, the FMG convergence path is shown for a â€œgradedâ€ set of tolerances. Specifically, for the 7-level cavity flow levels 2 through 6 were converged to -0.7, -1.3, -1.9, -2.5 and -3.1, respectively (factor of 1/4 reduction per level). These particular values are all equal to -2.4 if instead of Eq. 5.17, the residual is normed according to I Vi ,j I INI = E (5.19) aUi,jfluX' where flux is a characteristic momentum flux, equal to the Reynolds number in the present flow problem. Shyy and Sun [74] used this approach. The tolerance on level 6, -3.1, was chosen a posteriori to match the known initial level of the fine-grid residual. 140 The graded set of coarse-grid tolerances are representative of a â€œbest possible guessâ€ that one could make without prior experience. From these figures, there does not appear to be any benefit in converging the coarse grids to tighter tolerances. Furthermore, there is the disadvantage that tighter coarse-grid tolerances require more coarse-grid cycles and are therefore more expenÂ¬ sive in terms of work units and especially in terms of run time on the CM-5 (the bottom plot). The graded tolerances work almost as well as the truncation error criterion, except that there are a few unnecessary cycles on levels 2 and 3. The tradeoff between the run time elapsed during the FMG procedure on serial and parallel computers, and the initial level of the u residual, is summarized in Table 5.1. Coarse-grid tolerances Number of V(3,2) cycles on levels {1 ...6} FMG work units FMG CM-5 busy time Initial level of fine-grid U residual T.E. w/denom. = 1 {x 1 1 1 1 1} 2.2 2.3 s -3.25 T.E. w/denom. = 5 {x 2 3 6 6 5} 11.5 10.9 s -3.54 -3.0 on all levels {x 15 17 14 10 3} 11.1 20.2 s -3.46 -5.0 on all levels {x 24 27 26 30 32} 69.3 64.1 s -3.57 Graded tolerances {x 6 8 8 6 5} 11.9 13.0 s -3.54 Table 5.1. Comparison between different sets of coarse-grid tolerances in terms of the effort expended in the FMG procedure, for the Re â€” 5000 lid-driven cavity flow and the bilinear interpolation prolongation procedure. The defect-correction stabilization strategy is used. To judge which case is the â€œbest,â€ one asks how many work units or how much cpu time is required to reach a given level of the residual. A few fine-grid cycles are required to make up the difference in the initial levels of the fine-grid residual. These are charged at a rate of slightly more than 6.25 work units per V(3,2) cycle for this 7-level problem with the 321 x 321 fine grid, equivalent to about 1.5 seconds on a 128-VU CM-5. Thus, the â€œ1-FMGâ€ procedure (the first row) is judged to be the most efficient. 141 Evidently, the cavity flow problem is relatively benign in terms of the convection effect on the convergence rate characteristics. The truncation error estimate is imÂ¬ mediately satisfied on each of the coarse grids in the 7-level computation after only 1 V(3,2) cycle. Even less smoothing is possible for this problem, even though the Reynolds number is high. Table 5.2 clarifies the role of the FMG procedure in this flow problem. Number of V(3,2) cycles on levels {1 ...6} FMG work units FMG CM-5 busy time Initial level of fine-grid U residual {0 0 0 0 0 0} {x 0 0 0 0 0} {x 1 0 0 0 0} 0.006 â€” diverges â€” â€” diverges â€” 0.22 s -2.44 {x 1 1 0 0 0} 0.031 0.50 s -2.73 {x 1 1 1 0 0} 0.135 0.88 s -3.03 {x 1 1 1 1 0} 0.550 1.42 s -3.16 {x 1 1 1 1 1} 2.216 2.25 s -3.25 Table 5.2. Accuracy/effort tradeoff between a â€œ1-FMGâ€ approach (7th row), and simple V cycling with a zero initial guess on the fine grid (1st row). An approximate solution must be obtained on at least level 2 in order to avoid divergence for the 7-level Re â€” 5000 lid-driven cavity flow problem, when the relaxation factors are ujuv = u)c = 0.5. â€œFMG work unitsâ€ refers to the work units (proportional to a serial computerâ€™â€™s run time) already expended at the point when multigrid cycling on the finest grid level begins. â€œCM-5 busy timeâ€ is the corresponding measure of work on a 128-VU CM-5, in seconds. The â€œxâ€ in the column corresponding to level 1 means that 2 SIMPLE iterations were done on the coarsest grid. These data are for the defect-correction strategy. Thus, it is possible to prolong the solution directly from level 3, a 21 x 21 grid, to the fine grid. However, for the relaxation factors used, an initial guess on an even coarser grid (level 1 or 2) is not accurate enough to prevent the fine-grid V(3,2) cycles from diverging. The results in Figures 5.6-5.7 showed that the initial residual on the fine grid was independent of the degree of accuracy obtained on the coarser grid levels. Closer 142 examination shows that the initial residual levels on the coarse grid levels during the FMG procedure also do not appear to depend on the degree to which the next coarser grid level is converged. Furthermore this observation holds for second-order upwinding on all levels in the cavity flow, and for either defect-correction or second- order upwinding in the symmetric backward-facing step flow. The FMG convergence paths for the step flow, using second-order upwinding on all grid levels, are shown in Figures 5.9-5.12. There appears to be a certain maximum amount of accuracy that can be carÂ¬ ried over to the next finer grid with the bilinear interpolation prolongation. Since the truncation error convergence criterion does not exceed this amount of accuracy, and indeed the average residuals levels are virtually the same if the denominator in Eq. 5.16 is set to five, the results strongly suggest that the degree of accuracy on a given coarse grid which is exploitable is related to the differential error in the soluÂ¬ tion, i.e. the truncation error, and not the algebraic error. Thus, the results support the arguments made in the paragraph following Eq. 5.4. With regard to the performance of the truncation error criterion, the defect- correction and second-order upwind stabilization strategies showed similar results, in both flow problems. The initial fine-grid residual level and the stability of the subseÂ¬ quent multigrid iterations, however, appear to be strongly dependent on the convecÂ¬ tion schemes used. Table 5.3 summarizes the FMG convergence rates for second-order upwinding in the lid-driven cavity flow. The -3.0 and graded tolerance cases both converged with the defect-correction scheme, but with second-order upwinding they diverge. After several fine-grid cycles the -2.0 case diverges also. The difference between the cases is evidentâ€”many more coarse-grid cycles are taken in the cases which diverge. The source terms in the 143 Coarse-grid tolerances Number of V(3,2) cycles on levels {1 ...6} FMG work units FMG CM-5 busy time Initial level of fine-grid U residual T.E. w/denom. = 1 {x 1 2 2 2 5} 9.4 8.8 s -2.88 T.E. w/denom. - 5 {x 4 14 7 16 18} 37.7 40.0 s -3.50 -2.0 on all levels {x 35 22 19 6 1} 6.9 28.6 s -3.17 -3.0 on all levels {x 45 34 74 35 oo} diverges Graded tolerances {x 23 14 19 20 oo} diverges Table 5.3. Comparison between different sets of coarse-grid tolerances in terms of the effort expended in the FMG procedure, for the Re = 5000 lid-driven cavity flow and the bilinear interpolation prolongation procedure. The second-order upwind stabilization strategy is used. second-order upwind discretization appear to be a strong destabilizing factor in this flow problem. Furthermore, the fact that the -2.0 constant tolerance at least reaches the fine grid while the constant -3.0 tolerance diverges suggests that the amount of mismatch between the ending coarse-grid residual level and the beginning fine-grid residual level, which is greater for the -3.0 case than the -2.0 case, is related to the size of the destabilizing source terms in the initial fine-grid problem. Thus in addition to being wasteful of work units and/or cpu time, obtaining excessive accuracy on the coarse grids can actually be detrimental to the stability of multigrid iterations, depending on the discretization scheme. Evidently with relaxation factors uuv = uic = 0.5, second- order upwinding, with V(3,2) cycles and uu = uv = 3 and vc = 9 inner point-Jacobi inner iterations in each SIMPLE outer iteration, the Re = 5000 lid-driven cavity flow is difficult to solve. The multigrid iterations only converge for a relatively small range of coarse-grid tolerances. This range may be hard to find by trial and error. The truncation error criterion is useful in this regard. Similar observations are made for the symmetric backward-facing step flow. FigÂ¬ ures 5.9-5.12 are the corresponding results for the Re = 300 symmetric backwardÂ¬ facing step flow, using second-order upwinding on all coarse grid levels in the FMG 144 procedure. The convergence rate behavior of the second-order upwind scheme in the step flow is similar to the defect-correction scheme results in the lid-driven cavity flow. For the symmetric backward-facing step flow, a 321 x 81 fine grid with 5 multiÂ¬ grid levels was used. The coarsest grid was 21 x 6. As in the cavity flow cases, V(3,2) cycles were used with bilinear interpolation for the prolongation procedure and restriction procedure 3. The relaxation factors were uuv â€” 0.6 and uc = 0.4. As in previous cases, 3, 3, and 9 point-Jacobi inner iterations were used in each SIMPLE iteration for the u, u, and p' systems of equations, respectively. In Figure 5.9, the convergence path is similar to the cavity flow convergence pathâ€”except that in the cavity flow, the coarse-grid tolerances given by Eq. 5.15 were loose enough that only one cycle was needed on each of the coarse grids, yielding a â€œ1-FMGâ€ cycle. In the symmetric backward-facing step flow, more than one cycle is needed on each coarse-grid level to satisfy the truncation error criterion. The truncation error estimate (with the denominator set equal to three because the coarse- grid discretizations are second-order) converges on the following levels corresponding to the grid levels 2 to 4: -0.8, -2.9, -4.0. On the finest grid the estimated level is -4.9. Figures 5.10-5.12 show the FMG convergence path when tighter coarse-grid tolerÂ¬ ances are used, and these results are summarized in Table 5.4 below. For the graded set of coarse grid tolerances, levels 2 through 4 were converged to -3.1, -3.7, and -4.3. Each of these levels corresponds to the level -2.1 if the norm used is Eq. 5.19 instead of the average L\ norm. As in the cavity flow case, there is only a small effect on the initial solution accuracy on each coarse-grid level. There is no benefit to the initial fine-grid residual level by converging the coarse grids to strict tolerances. The truncation error criterion with the denominator set to 5 appears to be the most stringent criterion which does not waste any coarse-grid cycles, i.e. it is nearly the optimal cost/residual reduction 145 balance. The other approaches obtain more accuracy on the coarse grids than can be carried over to the initial fine-grid solution, for the bilinear interpolation prolongation. Coarse-grid tolerances Number of V(3,2) cycles on levels {1 ...4} FMG work units FMG CM-5 busy time Initial level of fine-grid U residual â€œ1-FMGâ€ cycle {x 1 1 1} 2.2 1.2 s -2.88 T.E. w/denom. = 1 {x 2 2 3} 5.9 3.0 s -3.82 T.E. w/denom. = 5 {x 4 4 4} 8.6 4.8 s -4.63 -3.0 on all levels {x 23 7 1} 6.5 8.1 s -4.37 -5.0 on all levels {x 45 16 10} 27.0 21.5 s -5.10 Graded tolerances {x 21 9 5} 13.8 10.8 s -4.71 Table 5.4. Comparison between different sets of coarse-grid tolerances in terms of the effort expended in the FMG procedure, for the Re â€” 300 symmetric backwardÂ¬ facing step flow and the bilinear interpolation prolongation procedure. Second-order upwinding is used on all grid levels. The results for the defect-correction strategy are summarized in the table below. In the cavity flow, the second-order upwind scheme was very difficult to converge when a constant or a graded tolerance was given. In the step flow, it appears that the defect-correction strategy is harder to converge. Coarse-grid tolerances Number of V(3,2) cycles on levels {1 - 4} FMG work units FMG CM-5 busy time Initial level of fine-grid U residual â€œ1-FMGâ€ cycle {x 1 1 1} 2.2 1.0 s -2.99 T.E. w/denom. - 1 {x 2 2 2} 4.3 1.9 s -3.34 T.E. w/denom. = 5 {x 5 6 5} 12.3 5.1 s -4.18 -3.0 on all levels {x 22 9 1} 7.3 6.9 s -4.00 -5.0 on all levels {x 32 24 53} 100.1 37.8 s -4.24 Graded tolerances {x 21 12 21} 41.4 17.2 s -4.22 Table 5.5. Comparison between different sets of coarse-grid tolerances in terms of the effort expended in the FMG procedure, for the Re = 300 symmetric backward-facing step flow and the bilinear interpolation prolongation procedure. The defect-correction stabilization strategy is used. 5.2.3 Influence of Initial Guess on Convergence Rate The cost/initial accuracy tradeoff was discussed above. In addition, the initial guess on the fine grid is important because it can affect the asymptotic convergence 146 rate and stability of subsequent fine-grid cycles. In many cases this consideration is more important than the cost/initial accuracy tradeoff, since the time spent in the FMG procedure may be very small compared to the overall time required if many fine-grid cycles are needed. The FMG contribution to the total run time, especially on the CM-5, is not always negligible, though, in particular if one defines converÂ¬ gence according to the truncation error estimate on the finest grid, i.e. differential convergence, as suggested by Brandt and Taâ€™asan [7]. Figure 5.13 gives the convergence path for the entire computation for the lid- driven cavity flow. In the top plot, the fine-grid average u residual is plotted against the CM-5 busy time for the defect-correction scheme. The defect-correction scheme and second-order upwind scheme (bottom plot) converge at nearly the same rate. The differences in the initial fine-grid residual level due to the FMG procedure evidently do not persist for very long, and if the purpose is to obtain algebraic convergence, Eq. 5.4, then the difference in CM-5 busy time due to the FMG procedure is insignifÂ¬ icant. However, if convergence is declared when the average u residual falls beneath the dotted line, the estimated truncation error level on the fine grid, then the FMG procedure contributes anywhere from 10% of the total time, in the case of the trunÂ¬ cation error criterion with denominator 1, to 80% of the total time, in the case of the constant -5.0 criterion. For the Re = 5000 lid-driven cavity flow, using SIMPLE with vu = vv = 1 and vc = 4 inner SLUR iterations and a W(l,l) multigrid cycle, Sockol [80] reported that 86 work units and 800 seconds on an Amdahl 5980 were needed to reach convergence. To reach a similar convergence tolerance the present computation needed 200 work units and 64 seconds on the CM-5. In the previous section, the amount of smoothing used in the present case, V(3,2) cycles, was observed to be somewhat more than was necessary for this flow problem. The difference between V(3,2) cycles and W(l,l) 147 cycles in terms of work units is approximately 3, per cycle. Thirty cycles on the fine grid were taken in the present case. Thus, it seems that the present result is comparable to Sockolâ€™s result. The fine-grid convergence paths for the symmetric backward-facing step flow, FigÂ¬ ure 5.14, are very interesting. The second-order upwind scheme performs remarkably well. The average u residual reaches -8.0 in just slightly more than 20 seconds on the CM-5 and 140 work units (20 V(3,2) cycles on the 321 x 81 fine grid). This convergence rate corresponds to an amplification factor of 0.6 per cycle for the L\ norm of the u-residual. Because of the fast convergence rate the contribution of the startup FMG cycling is a significant fraction of the overall parallel run time. The defect-correction strategy does not converge as quickly as the second-order upwind scheme in the symmetric backward-facing step flow. Furthermore, for the defect-correction scheme, the fine-grid initial guess evidently affects the rate of conÂ¬ vergence. To obtain the convergence paths in the top plot of Figure 5.14, identical procedures and parameters were used for the multigrid iterations beginning on the fine grid. The relaxation factors were ujuv = u>c = 0.5 and fixed V(3,2) cycles were used. The coarse-grid discretizations in the FMG procedure use first-order upwinding, while the fine-grid discretization is modified to produce central-difference accuracy. Thus, the sudden rise in the residual level for all cases (except the truncation erÂ¬ ror criterion with denominator equal to 1) suggests that the first-order upwind and central-difference solutions to this flow problem are very different. It is apparently difficult for the numerical method to evolve the solution from first-order upwind acÂ¬ curacy into central-difference accuracy. Thus, there is actually an advantage in not converging the coarse grids to tight tolerances. On the other hand, the â€œl-FMGâ€ procedure has the worst convergence rate of the cases considered. The conclusion 148 Figure 5.14 supports is that there is an optimal solution accuracy on the coarse grids in the FMG procedure, which is related to the differential error in the solution since the truncation error estimate gives the best result. 5.2.4 Remarks Both flow problems have strong nonlinearities and are relatively difficult and slow to converge as single-grid computations. The multigrid method allows larger relaxation parameters to be used. Very fast convergence rates can be obtained, but the performance depends on the discretization on coarse grids (the stabilization strategy) and the initial fine-grid guess. The fact that the truncation error criterion gives the best results in both flow problems, and that regardless of how tight the coarse grids are converged both the initial fine and coarse-grid residuals are relatively independent, indicates that there is only a certain amount of accuracy which can be obtained initially for a given flow problem and coarse-grid discretization scheme, and that this observation is essentially a reflection of the truncation error of the discretization. The second-order upwind scheme may be prone to large source terms which can cause the multigrid iterations to diverge, especially if relatively few smoothing iteraÂ¬ tions are used. This observation was made for the cavity flow. On the other hand, when there is a significant difference between the first-order and central-difference soÂ¬ lutions on a given grid, the success of the defect-correction strategy depends strongly on the initial guess on the finest grid (re: the step flow results) and, in this sense, the defect-correction approach is not very robust. The stability of multigrid iterations is different than for single-grid calculations, and certainly more confusing. For example, if a single-grid calculation does not conÂ¬ verge at a given Reynolds number with a certain set of relaxation parameters, then 149 reducing the relaxation factors is always convergence-enhancing. For multigrid iteraÂ¬ tions this is not necessarily true. It was observed for the second-order upwind scheme in the Re = 300 symmetric backward-facing step problem, that the single-grid method diverges using uuv = 0.3 with uc = 0.2 for the Re = 300 symmetric backward-facing step flow and the second-order upwind scheme. However, convergence was obtained with uuv = 0.6 and uc â€” 0.4. Evidently, there is a certain minimum amount of smoothing required. The amount depends on the flow problem as well as the restricÂ¬ tion and prolongation procedures. In other words, reducing the relaxation factors to cope w'ith problems that have strong nonlinearities may simultaneously require inÂ¬ creasing the number of smoothing iterations on each level. The converse is also true although perhaps counterintuitiveâ€”reducing the amount of smoothing, for example from V(3,2) to V(2,l) cycles, may cause stability problems. Increasing the relaxation factors is the appropriate response. By contrast, for single-grid computations, if the number of inner iterations is too low, the relaxation factors are decreased to avoid divergence. Additional testing in the smoothing/relaxation factor parameter space would be desirable to further clarify this point. 5.3 Performance on the CM-5 This section quantifies the cost of multigrid cycling on the CM-5, and discusses the efficiency and scalability of the present algorithm and implementation. In other words, to connect with the preceding section, once the fine-grid is reached, what is the best grid schedule to use, how long does each cycle take, and how does this cost scale with the problem size and the number of processors? In Figure 5.15, the costs of smoothing and prolongation are shown as a function of problem size, for a 32-node CM-5 and a 512-node CM-5. During a multigrid cycle these costs are incurred for each grid level. In a V(3,2) cycle, for example, 5 150 SIMPLE iterations are done at every grid level, along with one restriction from and one prolongation to every grid level except the coarsest. If the finest grid is 770 x 770, then on a 32-node CM-5 the subgrid size {VP) is roughly 4800. The next (coarser) grid is 385 x 385 and has a subgrid size of 1225. Thus in a two-level V(3,2) cycle, the total time is the sum of 5 SIMPLE iterations at VP = 4800, one restriction from VP = 4800 to VP = 1225, 5 SIMPLE iterations at VP = 1225, and one prolongation from VP = 1225 to VP â€” 4800. Thus, Figure 5.15 is a level-by-level breakdown of the parallel run time used by the smoothing and prolongation multigrid components. The times plotted are total elapsed times including the processor idle time due to front-end work. The smoothing cost dominates the cost of the prolongation, at every VP. Thus unless a multigrid cycle with less smoothing is used, the common idealization that the restriction and prolongation costs are negligible on serial computers also holds true on the CM-5. The restriction cost has not been shown in order to keep the figure clear. It follows the same trend as prolongation only slightly less time-consuming if the residuals are alone are restricted (about 25% less), and slightly more time-consuming if both solutions and residuals are restricted. The trend is linear for both restriction, prolongation and smoothing. When residÂ¬ uals only are restricted, the ratio of the times for these three components tends toward 1:2:13, on the 32-node CM-5, as the number of grid points increases (i.e. as the subgrid size increases). However, for the 512-node CM-5, the time taken by prolongation grows at a slightly greater rate than on the 32-node computer. On the 512-node CM-5, VP = 4800 corresponds to a 3080 x 3080 grid size, instead of 770 x 770 as was the case with the 32-node CM-5. Apparently, the global communication patterns needed to 151 accomplish the prolongation are not perfectly scalable on the fat-tree, at least with the current CM-Fortran implementation. Figure 5.15 gives the impression that the cost of SIMPLE iterations varies linearly with VP. However, as shown in Figure 5.16, the variation is not actually linear for very small VP. The bar on the left is the CM-5 busy time for 5 SIMPLE iterations, given as a function of the grid level. The bar on the right is the corresponding CM-5 elapsed time, taken from data points along the smoothing cost curve in Figure 5.15. The busy time records the time spent doing parallel computation and interprocessor communication operations. These operations are very inefficient at small VP on the CM-5 because the vector units are not fully loaded. Thus, the busy time does not scale linearly with the subgrid size for small VP because the efficiency of vectorized computation and interprocessor communication increases as the subgrid size grows. Note however that the busy time is always a monotonic function of VP. The variation of elapsed time by contrast stays approximately constant until level 5 of this sample multigrid cycle. Level 5 corresponds to VP â€” 36 on the 32-node CM-5. The elapsed time includes the idle-processor time due to front-end work. As discussed in Chapter 3, there are several overhead costs of parallel computation and interprocessor communication. These operations may leave the CM-5 vector units inactive for short periods of time. For small VP the dominant consideration in this regard is the passing of code blocks, i.e. the front-end-to-processor communication. This cost stays constant with VP, as shown for small the elapsed time at small VP in Figure 5.16. The elapsed time is actually larger for VP = 1 than VP = 2. This observation is reproducible but its cause is not fully understood. Inaccurate timings may be the problem. A computer with a relatively fast front-end and communication network performs closer to the ideal for small VP. 152 Since the cost of smoothing on the coarse grids does not go to zero as VP â€”> 0, the possibility exists for coarse grids to make a nonneglible contribution to the parallel run time, if the cycling scheme is such that the coarse grids are visited more frequently than the fine grids. Figures 5.17-5.18 illustrate this point clearly. The cost per multigrid cycle is compared between V and W cycling strategies. Specifically, V(3,2) cycles are compared against W(3,2) cycles. The timings are obtained on a 32-node CM-5. The number of levels is fixed as the finest grid dimensions increase. Both elapsed and busy times are plotted. The total time per cycle includes the cost of smoothing on the grid levels inÂ¬ volved, the restriction and prolongation costs, and the cost of program control and input/output. For a V cycle, this time can be modelled as (5.20) nlevel TimÃ©is) = "g'$t(n^ + ^ + Â£ {n + V CVde k=l k=2 where s*,, rk, and pk and the smoothing time per iteration on level k (from FigÂ¬ ure 5.15), the restriction time from level fc, and the prolongation time to level k. The number of levels is nÂ¡eiJeÂ¡ and npre and npost represent the number of pre and post-smoothing iterations, in this case 3 and 2, respectively. In contrast, W cycles visit the coarse grids much more frequently. Their time per cycle can be modelled 'TimpÃ Q ^ 71 level nlevel uy Y = Â£ sk(npre + npos<)2(ni'-'-fc) + Â£ (rk + pk)2(nâ€œÂ«'-*>. (5.21) These expressions are valid for serial computations, too. On serial computers, the restriction and prolongation costs are generally negligible, and the smoothing cost per level sk is basically a factor of 1/4 smaller for the lower (coarser) grid levels. For parallel computation on the CM-5, the fact that sk remains approximately constant for the coarsest grids is a problem when many multigrid levels are used. When only three levels are involved, there is very little disadvantage to using W cycles, as shown in Figure 5.17. Since it is usually possible to gain some benefit to the 153 convergence rate by more frequent coarse-grid corrections, W cycles are recommended on the CM-5 if the number of multigrid levels is small. However, for 5 or more levels, Figures 5.18 and 5.19, W cycles begin to cost more than they are worth in terms of improved convergence rates. Also, since there is a greater difference between V and W cycle elapsed and busy times as more multigrid levels are added, reflecting the relatively larger idle times for coarse grids (recall Figure 5.16), the parallel efficiency of W cycles is less than that of V cycles. In the present work V cycles have been sufficient to achieve good convergence rates so no comparisons have been made to W cycles. Such studies need to be made, but on a problem-by-problem basis. For the symmetric backward-facing step flow and lid-driven cavity flow, it is not expected that W cycles will be advantageous. In many cases it is acceptable and even beneficial to use less than the full compleÂ¬ ment of multigrid levels, i.e. to increase the problem size keeping the number of levels fixed. Whether or not the computation is for a physically time-dependent flow probÂ¬ lem, there exists an implied time-step in iterative numerical techniques. In multigrid computations, the changes in the evolving solution on coarser grid levels are smaller, reflecting the fact that the physical or pseudo-physical development of the solution on the fine-grid is occurring on a much smaller scale. Thus, the coarsest grid levels may be truncated without deteriorating the convergence rate. Pressure needs to be treated globally, but usually there are enough multigrid cycles taken to ensure that slow development of the pressure field is not a problem, even when the coarsest grid level is not very coarse. Figures 5.20-5.22 integrate the information contained in the preceding figures. In Figure 5.20, the variation of parallel efficiency of 7-level V(3,2) cycles with problem size is summarized. The problem size is the virtual processor ratio VP of the finest 154 grid level, but of course during the multigrid cycle operations are being done on coarser grids, too, where VP is smaller. Figure 5.20 is similar to Figure 3.2 obtained using the single-grid pressure-correction algorithm. For small VP the useful work (the computation) is dominated by the interprocessor and front-end-to-processor communication, resulting in low parallel efficiencies. The efficiency rises as the time spent in computation increases relative to the overhead costs. The highest efficiency obtained is almost 0.65 compared to 0.8 for the single-grid method on the CM-5. The burden of additional program control, relatively more expensive coarse-grid smoothing, and the restriction and prolongation tasks, adds up to 0.15 in terms of the parallel efficiency. Unlike the single-grid case however, the efficiency does not peak for large problem sizes. The contributions from the less-efficient coarser grids in a multigrid cycle on the CM-5 are significant, even when the finest grid has VP ~ 8k. The range of subgrid sizes comprising a 7-level multigrid cycle scale (a realistic cycle) span three orders of magnitude. Unfortunately, the range of UP in which the multigrid smoother achieves high parallel efficiencies is not as broad. In this regard the performance of the multigrid method on the MasPar-style of SIMD computers is expected to be much better since the single-grid method achieved high parallel efficiencies for VP > 32 all the way up to the largest problem size. Numerical experiments have not been conducted to study the multigrid method on MasPar SIMD computers, however, because their Fortran compiler is not yet sufficiently developed to address the storage problem. The efficiency in 5.20 apparently has a small dependence on the number of proÂ¬ cessors. This dependence is clearly shown in the next figure, Figure 5.21. The deÂ¬ pendence is due to the slightly increased time spent in intergrid transfer operations with increasing np, observed earlier in Figure 5.15. Figure 5.21 shows the decrease in 155 efficiency with increasing number of processors for five different subgrid sizes. Again recall that the subgrid size is for the finest grid but that much coarser grids are inÂ¬ volved in the 7-level V(3,2) cycles. The figure indicates that the rate of decrease in efficiency is the same for every VP down to at least VP â€” 320. The dashed lines are linear least-squares curve fits to the data. The data points are perturbed about these lines due to variations in the elapsed parallel run time Tv. Tp varies slightly from timing to timing depending on the workload of the front-end machine. In all cases multiple timings were obtained as a check on the reproducibility. In light front-end loadings (i.e. the middle of the night), the measured Tp did not vary more than +/-20%. Figure 5.22 is combines the information contained in Figures 5.20 and 5.21. As in the single-grid case, Figure 3.6, curves of constant efficiency are drawn on a plot of problem size versus the number of processors. The curves are constructed by interpolating in Figure 5.21, using the dashed lines as the data instead of the actual data points, to determine VP at a given (E,np) intersection. N is computed from the definition of VP, i.e. N = npVP. The isoefficiency curves are almost linear or, in other words, the 7-level multigrid algorithm analyzed on a per-cycle basis, is almost scalable. Each of the isoefficiency curves can be accommodated by an expression of the form N â€” No = constant (np â€” 32)Q, (5.22) with q ~ 1.1. The symbol No is the initial problem size needed to obtain a particular E on 32 processors. Along the isoefficiency curves, â€œscaled-speedupâ€™â€™ [35] is nearly achieved. If the parallel run time Tp at the initial problem size is acceptable, then it can be maintained with the present pressure-based multigrid method as the problem size and the number 156 of processors are increased in proportion. The inner iterations must be point-Jacobi of course, since the line-iterative method is 0(N log2 N). With the line-iterative method Tp increases slightly along the isoefficiency curves. The scalability should be nearly the same though, since nearest-neighbor communications dominate in the cyclic reduction parallel algorithm due to data-mapping used on the CM-5. 5.4 Concluding Remarks A parallel multigrid algorithm has been formulated and implemented on the CM- 5. The focus of numerical experiments and timings has been on the potential of this approach for the purpose of achieving scalable parallel computing techniques for application to the incompressible Navier-Stokes equations. The results obtained indicate that the efficiency of the parallel implementation of the nonlinear pressure-based multigrid method approaches 0.65 for large problem sizes, and is almost linearly scalable on the CM-5. The cost per V(3,2) cycle is about 1.5 s on a 128-vector unit CM-5 for a 7-level problem with a 321 x 321 fine grid. The cost per iteration is dominated by the smoothing cost, and thus much attention has been given to the details of the implementation and performance of the single-grid pressure-based method on SIMD computers. Restriction and prolongation are almost negligible, although they are responsible for the deviation from linear computational scalability observed in Figure 5.22. Very large problem sizes can be handled on the CM-5, up to 3074 x 3074 on a 32-node machine, provided the storage problem for Fortran multigrid implementations can be resolved. The speed of the multigrid code was not assessed directly, but reasonable estimates can be made based on the single-grid performance. For the single-grid SIMPLE method using the point-Jacobi solver, 417 MFlops was achieved on a 32-node (128 VU) CM-5. Since the multigrid cost per 7-level cycle is dominated by the smoothing 157 costs and the multigrid efficiency is 0.65 compared to 0.8 (about a 20% decrease), the speed is roughly 333 MFlops. Slightly improved efficiency and speed can be obtained with fewer multigrid levels. For unsteady flow calculations multigrid cycles with a small number of levels may perform reasonably well. This should be investigated. Several practical recommendations have been made regarding multigrid techÂ¬ niques for parallel computation. V cycles should be used unless the number of multigrid levels is small. W cycles are too expensive because due to the nonneg- ligible coarse-grid smoothing costs. The FMG procedure should be controlled by the truncation error estimate Eq. 5.16. The FMG procedure can affect not only the time needed to reach the fine grid, but also the asymptotic convergence rate and stability of multigrid iterations can be affected as well, as evident from FigÂ¬ ure 5.14. This observation may not carry over to the the locally-coupled explicit smoother. It should be tested in the same way. In terms of computational efficiency the locally-coupled explicit method has nearly the same properties on the CM-5 as the pressure-correction method, although the influence on the cost per iteration and efficiency from the coefficient computations is greater. Several algorithmic factors have been studied, in particular the coarse-grid disÂ¬ cretization (the stabilization strategy) and the restriction procedure are observed to be important to the multigrid convergence rate. It appears that the use of second- order upwinding on all grid levels and the restriction procedure 3, summing the residuals but not restricting the solutions, provides a very effective approach for both the symmetric backward-facing step flow and the lid-driven cavity flow. Smoothing rates per V(3,2) cycle of 0.6 can be maintained until the residual is driven down to the level of the roundoff error. The convergence rate with cell-face averaging for the restriction of solutions and residuals was considerably slower. Similar results were obtained for the cavity flow. 158 In terms of the coarse-grid discretization strategy, it appears that the popular defect-correction approach may not be as robust as the second-order upwinding strategy, at least for entering-type flow problems. In these types of flows, i.e. probÂ¬ lems with inflow and outflow, the proper formulation of the numerical method (the pressure-correction smoother) is critical for obtaining good convergence rates. Global mass conservation must be explicitly enforced during the course of iterations. Global mass conservation ensures that the system of pressure-correction equations has a solution, which is identified as an important prerequisite for obtaining reasonable convergence rates in open boundary problems. The well-posed numerical problem does not distinguish between inflow and outflow at the open boundaryâ€”if the nuÂ¬ merical treatement of the open boundary condition is reasonable and can induce convergence, the finite-volume staggered-grid pressure-correction method can obtain the correct numerical solution even if inflow occurs at a nominally outflow boundary. In conclusion, the results of this research indicate that pressure-based multigrid methods are computationally and numerically scalable algorithms on SIMD comÂ¬ puters. Taking proper account of the many implementational considerations, high parallel efficiencies can be achieved and maintained as the number of processors and the problem size increases. Likewise, the convergence rate dependence on problem size should be greatly decreased by the multigrid technique. Thus the present apÂ¬ proach is viable for massively-parallel numerical simulations of the incompressible Navier-Stokes equations, and should be developed further on SIMD computers. The target machine should be have fast nearest-neighbor and front-end-to-processor comÂ¬ munication compared to the speed of computation, so that reasonably high parallel 159 efficiencies can be obtained at small problem sizes. The knowledge and implementaÂ¬ tions gained in this research are immediately useful for exploiting the current comÂ¬ putational capabilities of the CM-5 and MP-2 SIMD computers, and are practical contributions which will facilitate future research in parallel CFD. 160 Figure 5.1. Schematic of an FMG V(3,2) multigrid cycle. 161 Re = 5000 Lid-Driven Cavity Flow Streamfunction U Velocity Component Vorticity Pressure Figure 5.2. Streamfunction, vorticity, and pressure contours for Re = 5000 lid-driven cavity flow, using the 2nd-order upwind convection scheme. The streamfunction contours are evenly spaced within the recirculation bubbles and in the interior of the flows, but this spacing is not the same. The actual velocities within the recirculation regions are relatively weak compared to the core flows. 162 Re = 300 Symmetric Backward-Facing Step Flow Streamfunction 012345678 U Velocity Component V Velocity Component Figure 5.3. Streamfunction, vorticity, pressure, and velocity component contours for Re = 300 symmetric backward-facing step flow, using the 2nd-order upwind convecÂ¬ tion scheme. The streamfunction contours are evenly spaced within the recirculation bubbles and in the interior of the flows, but this spacing is not the same. The actual velocities within the recirculation regions are relatively weak compared to the core flows. Initial MG-convergence path for T.E. criterion w/denom. = 1 163 CM-5 Busy Time (seconds) Figure 5.4. The convergence path of the u-residual norm during the FMG procedure for the Re = 5000 lid-driven cavity flow, using the defect-correction stabilization strategy. The truncation error criterion, with denominator 1, is used to determine the coarse-grid tolerances. The abscissas plot work units (proportional to a serial computerâ€™s cpu time), and CM-5 busy time. Initial MG-convergence path for T.E. criterion w/denom. = 5 164 Work Units CM-5 Busy Time (seconds) Figure 5.5. The convergence path of the u-residual norm during the FMG procedure for the Re = 5000 lid-driven cavity flow, using the defect-correction stabilization strategy. The truncation error criterion, with denominator 5, is used to determine the coarse-grid tolerances. The abscissas plot work units (proportional to a serial computerâ€™s cpu time), and CM-5 busy time. Initial MG-convergence path for constant -3.0 tolerances 165 CM-5 Busy Time (seconds) Figure 5.6. The convergence path of the u-residual norm during the FMG procedure for the Re = 5000 lid-driven cavity flow, using the defect-correction stabilization strategy. The coarse-grid convergence criterion is ||1| < â€”3.0 on every level. The abscissas plot work units (proportional to a serial computerâ€™s cpu time) and CM-5 busy time. Initial MG-convergence path for graded tolerances 166 Work Units CM-5 Busy Time (seconds) Figure 5.7. The convergence path of the u-residual norm during the FMG procedure for the Re = 5000 lid-driven cavity flow, using the defect-correction stabilization strategy. The coarse-grid convergence criteria are graded. For levels 2â€”6, |||| < â€”0.7,â€”1.3,â€”1.9,â€”2.5,â€”3.1. The abscissas plot work units (proportional to a serial computerâ€™s cpu time) and CM-5 busy time. 167 Initial MG-convergence path for T.E. criterion w/denom. = 1 Figure 5.8. The convergence path of the u-residual norm during the FMG procedure for the Re = 300 symmetric backward-facing step flow, with the defect-correction stabilization strategy. The truncation error criterion, with denominator 1, is applied to abbreviate coarse-grid multigrid cycling. 168 Initial MG-convergence path forT.E. criterion w/denom. = 1 CM-5 Busy Time (seconds) Figure 5.9. The convergence path of the u-residual norm during the FMG procedure for the Re = 300 symmetric backward-facing step flow, with second-order upwind- ing on all levels. The truncation error criterion, with denominator 1, is applied to abbreviate coarse-grid multigrid cycling. 169 Initial MG-convergence path for T.E. criterion w/denom. = 5 CM-5 Busy Time (seconds) Figure 5.10. The convergence path of the u-residual norm during the FMG procedure for the Re = 300 symmetric backward-facing step flow, using second-order upwind- ing on all levels. The truncation error criterion, with denominator 5, is applied to abbreviate coarse-grid multigrid cycling. 170 Initial MG-convergence path for constant -3.0 tolerances CM-5 Busy Time (seconds) Figure 5.11. The convergence path of the r/-residual norm during the FMG procedure for the Re = 300 symmetric backward-facing step flow, using second-order upwinding on all levels. The coarse-grid convergence criterion is ||rfc|| < â€”3.0 on every level. 171 Initial MG-convergence path for graded tolerances CM-5 Busy Time (seconds) Figure 5.12. The convergence path of the u-residual norm during the FMG procedure for the Re = 300 symmetric backward-facing step flow, using second-order upwinding on all levels. The coarse-grid convergence criteria are graded. For levels 2â€”4, || < -2.5,-3.1,-3.7. 172 Re = 5000 MG-Convergence Paths for different FMG procedures Figure 5.13. The convergence path of the average u-residual norm on the finest grid level in the 7-level Re â€” 5000 lid-driven cavity flow. The relaxation factors used were ujuv = uc = 0.5. 173 Re = 300 MG-Convergence Paths for different FMG procedures Figure 5.14. The convergence path of the average u-residual norm on the finest grid level in the 5-level Re = 300 symmetric backward-facing step flow. The relaxation factors used were ujuv = 0.6, and ujc = 0.4. 174 Relative Cost of Multigrid Components on 32 and 512 node CM-5s Virtual processor ratio, VP Figure 5.15. The relative cost of smoothing and prolongation per V-cycle, as a function of the problem size, for 32 and 512-node CM-5 computers (128 and 2048 processors, respectively). The run times are obtained from V(3,2) cycles, which have 5 smoothing iterations, 1 restriction, and 1 prolongation at each grid level. Elapsed time (includes front-end-to-processor communication) is plotted. The restriction cost is slightly less than the prolongation cost when only residuals are restricted, slightly more when solutions are restricted too, but the trend is the same as for prolongation and is therefore not shown for clarity. 175 Smoothing Costs by Level on a 32-node CM-5 1 2 6 15 36 153 561 2145 8385 Multigrid Level, and Virtual Processor Ratio Figure 5.16. Smoothing cost, in terms of elapsed and busy time on a 32-node CM-5, as a function of the multigrid level for a case with a 1024 x 1024 fine grid. The elapsed time is the one on the right (always greater than the busy time). The times correspond to one SIMPLE iteration. 176 3 Level V and W-Cycle Times on a 32-node CM-5 Virtual Processor Ratio, VP Figure 5.17. Parallel run time, per cycle, on a 32-node CM-5, as a function of the problem size. V(3,2) cycle cost is compared with W(3,2) cycle cost in terms of total elapsed time (dashed lines), and busy time (solid lines). As the problem size increases the number of multigrid levels remains fixed at three. 177 5 Level V and W-Cycle Times on a 32-node CM-5 Virtual Processor Ratio, VP Figure 5.18. Parallel run time, per cycle, on a 32-node CM-5, as a function of the problem size. V(3,2) cycle cost is compared with W(3,2) cycle cost in terms of total elapsed time (dashed lines), and busy time (solid lines). As the problem size increases the number of multigrid levels remains fixed at five. 178 7 Level V and W-Cycle Times on a 32-node CM-5 Virtual Processor Ratio, VP Figure 5.19. Parallel run time, per cycle, on a 32-node CM-5, as a function of the problem size. V(3,2) cycle cost is compared with W(3,2) cycle cost in terms of total elapsed time (dashed lines), and busy time (solid lines). As the problem size increases the number of multigrid levels remains fixed at seven. 179 Efficiency vs. Problem Size for 7-Level Multigrid using V(3,2) Cycles Figure 5.20. Parallel efficiency of the 7-level multigrid algorithm on the CM-5, as a function of the problem size. Efficiency is determined from Eq. 3.3, where Tp is the elapsed time for a fixed number of V(3,2) cycles and Ti is the parallel computation time (Tnode-cpu) multiplied by the number of processors. The trend is the same as for the single-grid algorithm, indicating the dominant contribution of the smoother to the overall multigrid cost. 180 Efficiency vs. Number of CM-5 Nodes for 7-Level Multigrid using V(3,2) Cycles 0.65 0.6 0.55 m 0.5 o S 0.45 'o = 0.35 co CL 0.3 0.25 0.2 Figure 5.21. Parallel efficiency of the 7-level multigrid algorithm on the CM-5, as a function of the number of processors, for several problem sizes. Efficiency is deÂ¬ termined from Eq. 3.3, where Tp is the elapsed time for a fixed number of V(3,2) cycles, and T\ is the parallel computation time (Tnode-cpu) multiplied by the number of processors. There is only a small fall-off in the efficiency as np increases. 1 1 o 1 â€”1 ~o~ VP = 6400 o ~ ~ â€” â€” â€” __ a -o ~ ~ ~ ~ T)~ _____ VP = 3300 p - ~ ~ ~ X)~ _ _ _ _ VP = 2100 -O - Â°- -G- â€” â€” â€” o co 00 ll CL I >1 / 1 1 1 0 - o a _? - - ~ ~ ~ ~ - - - - _ _ _ VP = 320 o - l l o 1 1 32 64 256 512 Number of CM-5 Nodes 181 Isoefficiency Curves For 7-Level Multigrid using V(3,2) Cycles Number of CM-5 Nodes Figure 5.22. Isoefficiency curves for the 7-level pressure-correction multgrid method, based on timings of a fixed number of V(3,2) cycles, using point-Jacobi inner itÂ¬ erations. The plot is constructed based on linear least-squares curve fits of the data in Figures 5.21 and 5.20. The isoefficiency curves have the general form N = QUp + constant, where (3 ~ 1.1 for the efficiencies shown. REFERENCES [1] W. F. Ames. Numerical Methods for Partial Differential Equations. Computer Science and Applied Mathematics. Academic Press, San Diego, second edition, 1977. [2] J. B. Bell. P. 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SIAM Journal of Scientific and Statistical Computing, 6(2):492-503, April 1985. [94] S. Zeng and P. Wesseling. Numerical study of a multigrid method with four smoothing methods for the incompressible Navier-Stokes equations in general coordinates. In S. McCormick, editor, Proceedings of the Third Copper Mountain Conference on Multigrid Methods. Marcel Dekker, New York, 1993. BIOGRAPHICAL SKETCH Edwin Blosch received his B.S. degree with high honors from the University of Florida in 1989, received his M.S. degree, also from U.F., in 1991, and anticipates receiving his Ph.D. degree in 1994. He wants to be the first person to do a complete numerical simulation of a practically important physical process, for example meÂ¬ teorological or oceanographic particle transport, combustion, or the manufacturing processes of alloys, with as little modelling as possible, and enough space and time resolution so that the public will have no trouble recognizing the utility of his work and of scientific computing in general. Away from work he enjoys golf, basketball, and travelling with his wife. 189 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Wei Shyy, Chairman " â€˜ Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Chen-Chi Hsu Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Bruce Carroll Associate Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in to acceptable standards of scholarly presentation ahd scope and quality, as a dissertation for the degree pf D< opinion it conforms |s fully adequate, in ctor of Philosophy. David Mikolaitis Associate Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Sartaj Sahni Professor of Computer and Information Sciences This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulÂ¬ fillment of the requirements for the degree of Doctor of Philosophy. December 1994 Â£ Winfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School LD 1780 ,1991 Â»0 UNIVERSITY OF FLORIDA 3 1262 08553 7057 |